Post on 26-Jul-2020
MATH IN CAREERS
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Jewelry Maker A jewelry maker designs and creates jewelry Jewelry makers often employ geometric designs and shapes in their work and so they need a good understanding of geometry For example they must calculate volume and surface area to determine the amount of materials needed They can also use computer designing programs to help them with their design specifications Jewelry makers often need to calculate costs of materials and labor to determine production costs for their designs
If you are interested in a career as a jewelry maker you should study these mathematical subjects
bull Algebrabull Geometrybull Business Math
Research other careers that require knowing the geometry of three-dimensional objects Check out the career activity at the end of the unit to find out how jewelry makers use math
VolumeUNIT 9
MODULE 21Volume Formulas
Unit 9 1117
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Reading Start -Up
Visualize VocabularyUse the words and draw examples to complete the chart
Object Example
Understand VocabularyComplete the sentences using the preview words
1 A cone whose axis is perpendicular to its base is called a(n)
2 A prism that has at least one nonrectangular lateral face is called a(n)
Active Reading
Pyramid Create a Pyramid and organize the adjectives used to describe different objectsmdashright regular obliquemdashon each of its faces When listening to descriptions of objects look for these words and associate them with the object that follows
VocabularyReview Words area (aacuterea) composite figure (figura compuesta)
cone (cono)
cylinder (cilindro)
pyramid (piraacutemide)
sphere (esfera) volume (volume)
Preview Wordsapothem (apotema)oblique cylinder
(cilindro oblicuo)oblique prism
(prisma oblicuo)regular pyramid
(piraacutemide regular)right cone (cono recto)right cylinder (cilindro recto)right prism (prisma recto)
1118Unit 9
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MODULEcopy
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Stri
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Cor
bis
REAL WORLD VIDEO Check out how volume formulas can be used to find the volumes of real-world objects including sinkholes
Volume Formulas
MODULE PERFORMANCE TASK PREVIEW
How Big Is That SinkholeIn 2010 a giant sinkhole opened up in a neighborhood in Guatemala and swallowed up the three-story building that stood above it In this module you will choose and apply an appropriate formula to determine the volume of this giant sinkhole
Essential Question How can you use volume formulas to solve real-world problems
21LESSON 211
Volume of Prisms and Cylinders
LESSON 212
Volume of Pyramids
LESSON 213
Volume of Cones
LESSON 214
Volume of Spheres
LESSON 215
Scale Factor
Module 21 1119
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YOUAre Readycopy
Hou
ghton M
ifflin Harcourt Pub
lishing C
omp
any
bull Online Homeworkbull Hints and Helpbull Extra Practice
Write the equation for the area of a circle of radius r
Substitute the radius
Simplify
Complete these exercises to review skills you will need for this module
Area of a CircleExample 1 Find the area of a circle with radius equal to 5
A = π r 2
A = π (5) 2 A = 25π
Find each area
1 A circle with radius 4 2 A circle with radius 6
3 A circle with radius 3π 4 A circle with radius 2 _ π
Volume PropertiesExample 2 Find the number of cubes that are 1 cm 3 in size that fit into a cube of size 1 m 3
Notice that the base has a length and width of 1 m or 100 cm so its area is 1 m 2 or 10000 cm 2
The 1 m 3 cube is 1 m or 100 cm high so multiply the area of the base by the height to find the volume of 1000000 cm 3
Find the volume
5 The volume of a 1 km 3 body of water in m 3
6 The volume of a 1 ft 3 box in in 3
Volume of Rectangular PrismsExample 3 Find the volume of a rectangular prism with height 4 cm length 3 cm and width 5 cm
V = Bh
V = (3) (5) (4)
V = 60 cm 3
Find each volume
7 A rectangular prism with length 3 m width 4 m and height 7 m
8 A rectangular prism with length 2 cm width 5 cm and height 12 cm
Write the equation for the volume of a rectangular prism
The volume of a rectangular prism is the area of the base times the height
Simplify
Module 21 1120
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A Bh
C
b
ℓ
A
B
C
D
A
B
C
D
E
F
R
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Name Class Date
Explore Developing a Volume FormulaAs shown at the left below
_ AB has length b and C is any point on line ℓ parallel to
_ AB The distance between the
line containing _ AB and line ℓ is h No matter where C is located on line ℓ the area of the resulting ABC is always
a constant equal to 1 __ 2 bh Similarly given a polygon and a plane R that is parallel to the plane containing the polygon suppose you choose a point on R and create a pyramid with the chosen point as the vertex and the polygon as the base Both the base area and the height of the pyramid remain constant as you vary the location of the vertex on R so it is reasonable to assume that the volume of the pyramid remains constant
Postulate
Pyramids that have equal base areas and equal heights have equal volumes
Consider a triangular pyramid with vertex A directly over vertex D of the base BCD This triangular pyramid A-BCD can be thought of as part of a triangular prism with EFA cong BCD Let the area of the base be B and let AD = h
A What is the volume of the triangular prism
Resource Locker
Module 21 1133 Lesson 2
212 Volume of PyramidsEssential Question How do you find the volume of a pyramid
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A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
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B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
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h
B1
B2
B3B4
16 cm
24 cm
24 cm
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
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15 cm25 cm
15 cm
146 m
230 m230 m
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Mark
Go
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ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
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15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
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5 ft
3 ft
10 ft
2 ft
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Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
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A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
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12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
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10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
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4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
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10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
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Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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oug
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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oug
hton Mifflin H
arcourt Publishin
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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hton Mifflin H
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
Reading Start -Up
Visualize VocabularyUse the words and draw examples to complete the chart
Object Example
Understand VocabularyComplete the sentences using the preview words
1 A cone whose axis is perpendicular to its base is called a(n)
2 A prism that has at least one nonrectangular lateral face is called a(n)
Active Reading
Pyramid Create a Pyramid and organize the adjectives used to describe different objectsmdashright regular obliquemdashon each of its faces When listening to descriptions of objects look for these words and associate them with the object that follows
VocabularyReview Words area (aacuterea) composite figure (figura compuesta)
cone (cono)
cylinder (cilindro)
pyramid (piraacutemide)
sphere (esfera) volume (volume)
Preview Wordsapothem (apotema)oblique cylinder
(cilindro oblicuo)oblique prism
(prisma oblicuo)regular pyramid
(piraacutemide regular)right cone (cono recto)right cylinder (cilindro recto)right prism (prisma recto)
1118Unit 9
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ACA-A
MODULEcopy
Hou
ght
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iffl
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bull Im
age
Cre
dit
s copy
Stri
ng
er
Reut
ers
Cor
bis
REAL WORLD VIDEO Check out how volume formulas can be used to find the volumes of real-world objects including sinkholes
Volume Formulas
MODULE PERFORMANCE TASK PREVIEW
How Big Is That SinkholeIn 2010 a giant sinkhole opened up in a neighborhood in Guatemala and swallowed up the three-story building that stood above it In this module you will choose and apply an appropriate formula to determine the volume of this giant sinkhole
Essential Question How can you use volume formulas to solve real-world problems
21LESSON 211
Volume of Prisms and Cylinders
LESSON 212
Volume of Pyramids
LESSON 213
Volume of Cones
LESSON 214
Volume of Spheres
LESSON 215
Scale Factor
Module 21 1119
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-CCA-C
YOUAre Readycopy
Hou
ghton M
ifflin Harcourt Pub
lishing C
omp
any
bull Online Homeworkbull Hints and Helpbull Extra Practice
Write the equation for the area of a circle of radius r
Substitute the radius
Simplify
Complete these exercises to review skills you will need for this module
Area of a CircleExample 1 Find the area of a circle with radius equal to 5
A = π r 2
A = π (5) 2 A = 25π
Find each area
1 A circle with radius 4 2 A circle with radius 6
3 A circle with radius 3π 4 A circle with radius 2 _ π
Volume PropertiesExample 2 Find the number of cubes that are 1 cm 3 in size that fit into a cube of size 1 m 3
Notice that the base has a length and width of 1 m or 100 cm so its area is 1 m 2 or 10000 cm 2
The 1 m 3 cube is 1 m or 100 cm high so multiply the area of the base by the height to find the volume of 1000000 cm 3
Find the volume
5 The volume of a 1 km 3 body of water in m 3
6 The volume of a 1 ft 3 box in in 3
Volume of Rectangular PrismsExample 3 Find the volume of a rectangular prism with height 4 cm length 3 cm and width 5 cm
V = Bh
V = (3) (5) (4)
V = 60 cm 3
Find each volume
7 A rectangular prism with length 3 m width 4 m and height 7 m
8 A rectangular prism with length 2 cm width 5 cm and height 12 cm
Write the equation for the volume of a rectangular prism
The volume of a rectangular prism is the area of the base times the height
Simplify
Module 21 1120
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-CCA-C
A Bh
C
b
ℓ
A
B
C
D
A
B
C
D
E
F
R
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaAs shown at the left below
_ AB has length b and C is any point on line ℓ parallel to
_ AB The distance between the
line containing _ AB and line ℓ is h No matter where C is located on line ℓ the area of the resulting ABC is always
a constant equal to 1 __ 2 bh Similarly given a polygon and a plane R that is parallel to the plane containing the polygon suppose you choose a point on R and create a pyramid with the chosen point as the vertex and the polygon as the base Both the base area and the height of the pyramid remain constant as you vary the location of the vertex on R so it is reasonable to assume that the volume of the pyramid remains constant
Postulate
Pyramids that have equal base areas and equal heights have equal volumes
Consider a triangular pyramid with vertex A directly over vertex D of the base BCD This triangular pyramid A-BCD can be thought of as part of a triangular prism with EFA cong BCD Let the area of the base be B and let AD = h
A What is the volume of the triangular prism
Resource Locker
Module 21 1133 Lesson 2
212 Volume of PyramidsEssential Question How do you find the volume of a pyramid
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A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
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oug
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pany
B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
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h
B1
B2
B3B4
16 cm
24 cm
24 cm
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
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15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
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oug
hton Mifflin H
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pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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arcourt Publishin
g Com
pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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cour
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pan
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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oug
hton Mifflin H
arcourt Publishin
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pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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oug
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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oug
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g Com
pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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pan
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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cour
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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oug
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cour
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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arcourt Publishin
g Com
pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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oug
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cour
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lishi
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pan
y
9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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cour
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pan
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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arcourt Publishin
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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oug
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Mif
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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oug
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arcourt Publishin
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pany
Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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oug
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cour
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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oug
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arcourt Publishin
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pany
C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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oug
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cour
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pan
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
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red
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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hton Mifflin H
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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pan
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
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g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
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cour
t Pub
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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Mif
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cour
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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MODULEcopy
Hou
ght
on M
iffl
in H
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ublis
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omp
any
bull Im
age
Cre
dit
s copy
Stri
ng
er
Reut
ers
Cor
bis
REAL WORLD VIDEO Check out how volume formulas can be used to find the volumes of real-world objects including sinkholes
Volume Formulas
MODULE PERFORMANCE TASK PREVIEW
How Big Is That SinkholeIn 2010 a giant sinkhole opened up in a neighborhood in Guatemala and swallowed up the three-story building that stood above it In this module you will choose and apply an appropriate formula to determine the volume of this giant sinkhole
Essential Question How can you use volume formulas to solve real-world problems
21LESSON 211
Volume of Prisms and Cylinders
LESSON 212
Volume of Pyramids
LESSON 213
Volume of Cones
LESSON 214
Volume of Spheres
LESSON 215
Scale Factor
Module 21 1119
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YOUAre Readycopy
Hou
ghton M
ifflin Harcourt Pub
lishing C
omp
any
bull Online Homeworkbull Hints and Helpbull Extra Practice
Write the equation for the area of a circle of radius r
Substitute the radius
Simplify
Complete these exercises to review skills you will need for this module
Area of a CircleExample 1 Find the area of a circle with radius equal to 5
A = π r 2
A = π (5) 2 A = 25π
Find each area
1 A circle with radius 4 2 A circle with radius 6
3 A circle with radius 3π 4 A circle with radius 2 _ π
Volume PropertiesExample 2 Find the number of cubes that are 1 cm 3 in size that fit into a cube of size 1 m 3
Notice that the base has a length and width of 1 m or 100 cm so its area is 1 m 2 or 10000 cm 2
The 1 m 3 cube is 1 m or 100 cm high so multiply the area of the base by the height to find the volume of 1000000 cm 3
Find the volume
5 The volume of a 1 km 3 body of water in m 3
6 The volume of a 1 ft 3 box in in 3
Volume of Rectangular PrismsExample 3 Find the volume of a rectangular prism with height 4 cm length 3 cm and width 5 cm
V = Bh
V = (3) (5) (4)
V = 60 cm 3
Find each volume
7 A rectangular prism with length 3 m width 4 m and height 7 m
8 A rectangular prism with length 2 cm width 5 cm and height 12 cm
Write the equation for the volume of a rectangular prism
The volume of a rectangular prism is the area of the base times the height
Simplify
Module 21 1120
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A Bh
C
b
ℓ
A
B
C
D
A
B
C
D
E
F
R
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaAs shown at the left below
_ AB has length b and C is any point on line ℓ parallel to
_ AB The distance between the
line containing _ AB and line ℓ is h No matter where C is located on line ℓ the area of the resulting ABC is always
a constant equal to 1 __ 2 bh Similarly given a polygon and a plane R that is parallel to the plane containing the polygon suppose you choose a point on R and create a pyramid with the chosen point as the vertex and the polygon as the base Both the base area and the height of the pyramid remain constant as you vary the location of the vertex on R so it is reasonable to assume that the volume of the pyramid remains constant
Postulate
Pyramids that have equal base areas and equal heights have equal volumes
Consider a triangular pyramid with vertex A directly over vertex D of the base BCD This triangular pyramid A-BCD can be thought of as part of a triangular prism with EFA cong BCD Let the area of the base be B and let AD = h
A What is the volume of the triangular prism
Resource Locker
Module 21 1133 Lesson 2
212 Volume of PyramidsEssential Question How do you find the volume of a pyramid
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
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oug
hton Mifflin H
arcourt Publishin
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pany
B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
h
B1
B2
B3B4
16 cm
24 cm
24 cm
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
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pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Com
pan
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
copy H
oug
hton Mifflin H
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pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
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A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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oug
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
copy H
oug
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arcourt Publishin
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pany
Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
copy H
oug
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arcourt Publishin
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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cour
t Pub
lishi
ng
Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany bull Im
age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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Com
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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hton Mifflin H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
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g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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hton Mifflin H
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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Har
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
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Har
cour
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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cour
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
YOUAre Readycopy
Hou
ghton M
ifflin Harcourt Pub
lishing C
omp
any
bull Online Homeworkbull Hints and Helpbull Extra Practice
Write the equation for the area of a circle of radius r
Substitute the radius
Simplify
Complete these exercises to review skills you will need for this module
Area of a CircleExample 1 Find the area of a circle with radius equal to 5
A = π r 2
A = π (5) 2 A = 25π
Find each area
1 A circle with radius 4 2 A circle with radius 6
3 A circle with radius 3π 4 A circle with radius 2 _ π
Volume PropertiesExample 2 Find the number of cubes that are 1 cm 3 in size that fit into a cube of size 1 m 3
Notice that the base has a length and width of 1 m or 100 cm so its area is 1 m 2 or 10000 cm 2
The 1 m 3 cube is 1 m or 100 cm high so multiply the area of the base by the height to find the volume of 1000000 cm 3
Find the volume
5 The volume of a 1 km 3 body of water in m 3
6 The volume of a 1 ft 3 box in in 3
Volume of Rectangular PrismsExample 3 Find the volume of a rectangular prism with height 4 cm length 3 cm and width 5 cm
V = Bh
V = (3) (5) (4)
V = 60 cm 3
Find each volume
7 A rectangular prism with length 3 m width 4 m and height 7 m
8 A rectangular prism with length 2 cm width 5 cm and height 12 cm
Write the equation for the volume of a rectangular prism
The volume of a rectangular prism is the area of the base times the height
Simplify
Module 21 1120
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A Bh
C
b
ℓ
A
B
C
D
A
B
C
D
E
F
R
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oug
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cour
t Pub
lishi
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Com
pan
y
Name Class Date
Explore Developing a Volume FormulaAs shown at the left below
_ AB has length b and C is any point on line ℓ parallel to
_ AB The distance between the
line containing _ AB and line ℓ is h No matter where C is located on line ℓ the area of the resulting ABC is always
a constant equal to 1 __ 2 bh Similarly given a polygon and a plane R that is parallel to the plane containing the polygon suppose you choose a point on R and create a pyramid with the chosen point as the vertex and the polygon as the base Both the base area and the height of the pyramid remain constant as you vary the location of the vertex on R so it is reasonable to assume that the volume of the pyramid remains constant
Postulate
Pyramids that have equal base areas and equal heights have equal volumes
Consider a triangular pyramid with vertex A directly over vertex D of the base BCD This triangular pyramid A-BCD can be thought of as part of a triangular prism with EFA cong BCD Let the area of the base be B and let AD = h
A What is the volume of the triangular prism
Resource Locker
Module 21 1133 Lesson 2
212 Volume of PyramidsEssential Question How do you find the volume of a pyramid
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A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
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h
B1
B2
B3B4
16 cm
24 cm
24 cm
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
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pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
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15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
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5 ft
3 ft
10 ft
2 ft
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oug
hton Mifflin H
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g Com
pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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cour
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lishi
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Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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arcourt Publishin
g Com
pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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cour
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Com
pan
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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oug
hton Mifflin H
arcourt Publishin
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pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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oug
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Mif
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cour
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pan
y
HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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oug
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Mif
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Har
cour
t Pub
lishi
ng
Com
pan
y
Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
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18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
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19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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otog
raphyIm
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ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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hton Mifflin H
arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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cour
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
A Bh
C
b
ℓ
A
B
C
D
A
B
C
D
E
F
R
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oug
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pan
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Name Class Date
Explore Developing a Volume FormulaAs shown at the left below
_ AB has length b and C is any point on line ℓ parallel to
_ AB The distance between the
line containing _ AB and line ℓ is h No matter where C is located on line ℓ the area of the resulting ABC is always
a constant equal to 1 __ 2 bh Similarly given a polygon and a plane R that is parallel to the plane containing the polygon suppose you choose a point on R and create a pyramid with the chosen point as the vertex and the polygon as the base Both the base area and the height of the pyramid remain constant as you vary the location of the vertex on R so it is reasonable to assume that the volume of the pyramid remains constant
Postulate
Pyramids that have equal base areas and equal heights have equal volumes
Consider a triangular pyramid with vertex A directly over vertex D of the base BCD This triangular pyramid A-BCD can be thought of as part of a triangular prism with EFA cong BCD Let the area of the base be B and let AD = h
A What is the volume of the triangular prism
Resource Locker
Module 21 1133 Lesson 2
212 Volume of PyramidsEssential Question How do you find the volume of a pyramid
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
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oug
hton Mifflin H
arcourt Publishin
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pany
B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
h
B1
B2
B3B4
16 cm
24 cm
24 cm
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
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arcourt Publishin
g Com
pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
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15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
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oug
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pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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oug
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
copy H
oug
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arcourt Publishin
g Com
pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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oug
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Mif
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cour
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lishi
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Com
pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
copy H
oug
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arcourt Publishin
g Com
pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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oug
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cour
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lishi
ng
Com
pan
y
9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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oug
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cour
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pan
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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arcourt Publishin
g Com
pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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oug
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Mif
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cour
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lishi
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Com
pan
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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oug
hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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hton Mifflin H
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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Mif
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cour
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
A
B
C
D
E
F
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
B Draw _ EC on one face of the triangular prism Consider the three pyramids A-BCD A-EBC
and A-CFE Explain why the sum of the volumes of these three pyramids is equal to the volume of the prism
C _ EC is the diagonal of a rectangle so EBC cong CFE
Explain why pyramids A-EBC and A-CFE have the same volume Explain why pyramids C-EFA and A-BCD have the same volume
D A-CFE and C-EFA are two names for the same pyramid so you now have shown that the three pyramids that form the triangular prism all have equal volume Compare the volume of the pyramid A-BCD and the volume of the triangular prism Write the volume of pyramid A-BCD in terms of B and h
Module 21 1134 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
h
B1
B2
B3B4
16 cm
24 cm
24 cm
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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pany bull Im
age C
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Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
15 ft
12 ft25 ft
30 cm12 cm
15 cm
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cour
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Com
pan
y
Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
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oug
hton Mifflin H
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g Com
pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
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A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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oug
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
copy H
oug
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arcourt Publishin
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pany
Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
copy H
oug
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arcourt Publishin
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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cour
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lishi
ng
Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany bull Im
age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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hton Mifflin H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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oug
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
copy H
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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hton Mifflin H
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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h
B1
B2
B3B4
16 cm
24 cm
24 cm
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oug
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pan
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Reflect
1 Explain how you know that the three pyramids that form the triangular prism all have the same volume
Explain 1 Finding the Volume of a Pyramid In the Explore you showed that the volume of a ldquowedge pyramidrdquo having its vertex directly over one of the vertices of the base is one-third the product of the base area and the height Now consider a general pyramid As shown in the figure a pyramid can be partitioned into nonoverlapping wedge pyramids by drawing a perpendicular from the vertex to the base The volume V of the given pyramid is the sum of the volumes of the wedge pyramids
That is V = 1 __ 3 B 1 h + 1 __ 3 B 2 h + 1 __ 3 B 3 h + 1 __ 3 B 4 h
Using the distributive property this may be rewritten as V = 1 __ 3 h ( B 1 + B 2 + B 3 + B 4 ) Notice that B 1 + B 2 + B 3 + B 4 = B where B is the base area of the given pyramid
So V = 1 __ 3 Bh
The above argument provides an informal justification for the following result
Volume of a Pyramid
The volume V of a pyramid with base area B and height h is given by V= 1 __ 3 Bh
Example 1 Solve a volume problem
A Ashton built a model square-pyramid with the dimensions shown What is the volume of the pyramid
The pyramid is composed of wooden blocks that are in the shape of cubes A block has the dimensions 4 cm by 4 by 4 cm How many wooden blocks did Ashton use to build the pyramid
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 24 cm So B = 576 cm 2
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3 576 16
So V = 3072 c m 3
bull Find the volume of an average block
The volume of a cube is given by the formula V = s 3 So the volume W of a wooden block is 64 c m 3
bull Find the approximate number of wooden blocks in the pyramid divide V by W So the number of blocks that Ashton used is 48
Module 21 1135 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
hton Mifflin H
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pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
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15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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oug
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g Com
pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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cour
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pan
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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pan
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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cour
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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pan
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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hton Mifflin H
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otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Com
pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
15 cm25 cm
15 cm
146 m
230 m230 m
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oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
Mark
Go
dd
ardiSto
ckPhotocom
B The Great Pyramid in Giza Egypt is approximately a square pyramid with the dimensions shown The pyramid is composed of stone blocks that are rectangular prisms An average block has dimensions 13 m by 13 m by 07 m Approximately how many stone blocks were used to build the pyramid Round to the nearest hundred thousand
bull Find the volume of the pyramid
The area of the base B is the area of the square with sides of length 230 m So
The volume V of the pyramid is 1 _ 3 Bh = 1 _ 3
So V asymp
bull Find the volume of an average block
The volume of a rectangular prism is given by the formula So the volume W of an
average block is
bull Find the approximate number of stone blocks in the pyramid divide
by So the approximate number of blocks is
Reflect
2 What aspects of the model in Part B may lead to inaccuracies in your estimate
3 Suppose you are told that the average height of a stone block 069 m rather than 07 m Would the increase or decrease your estimate of the total number of blocks in the pyramid Explain
Your Turn
4 A piece of pure silver in the shape of a rectangular pyramid with the dimensions shown has a mass of 197 grams What is the density of silver Round to the nearest tenth (Hint density = mass _____ volume )
Module 21 1136 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
15 ft
12 ft25 ft
30 cm12 cm
15 cm
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oug
hton
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t Pub
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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pan
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
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hton Mifflin H
arcourt Publishin
g Com
pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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pan
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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cour
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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oug
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cour
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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oug
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cour
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pan
y
9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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oug
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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arcourt Publishin
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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oug
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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oug
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cour
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Com
pan
y
Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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oug
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pany
C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
copy H
oug
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flin
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cour
t Pub
lishi
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Com
pan
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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pan
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
copy H
oug
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arcourt Publishin
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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Mif
flin
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cour
t Pub
lishi
ng
Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
15 ft
12 ft25 ft
30 cm12 cm
15 cm
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Explain 2 Finding the Volume of a Composite FigureYou can add or subtract to find the volume of composite figures
Example 2 Find the volume of the composite figure formed by a pyramid removed from a prism Round to the nearest tenth
A
bull Find the volume of the prismV = lwh = (25) (12) ( 15 ) = 4500 ft 3
bull Find the volume of pyramid Area of base B = (25) ( 12 ) = 300 ft 2
Volume of pyramid V = 1 _ 3 (300) (15) = 1500 ft 3
bull Subtract the volume of the pyramid from volume of the prism to find the volume of the composite figure
4500 - 1500 = 3000
So the volume of the composite figure is 3000 ft 3
B
bull Find the volume of the prism
V = lwh = (30) ( ) ( ) = ( ) cm 3
bull Find the volume of the pyramid
Area of base B = cm 2
Volume of pyramid V = 1 _ 3 ( ) ( ) = ( ) c m 3
bull Subtract volume of pyramid from volume of prism to find volume of composite figure
- =
So the volume of the composite figure is c m 3
Module 21 1137 Lesson 2
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5 ft
3 ft
10 ft
2 ft
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pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
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A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
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4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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y
20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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hton Mifflin H
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
5 ft
3 ft
10 ft
2 ft
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pany
Your Turn
Find the volume of the composite figure Round to the nearest tenth
5 The composite figure is formed from two pyramids The base of each pyramid is a square with a side length of 6 inches and each pyramid has a height of 8 inches
6 The composite figure is formed by a rectangular prism with two square pyramids on top of it
Elaborate
7 Explain how the volume of a pyramid is related to the volume of a prism with the same base and height
8 If the length and width of a rectangular pyramid are doubled and the height stays the same how does the volume of the pyramid change Explain
9 Essential Question Check-In How do you calculate the volume of a pyramid
Module 21 1138 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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Mif
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cour
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lishi
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Com
pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
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12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
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10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
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10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
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Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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oug
hton Mifflin H
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
hton Mifflin H
arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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oug
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
arcourt Publishin
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pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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oug
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cour
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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cour
t Pub
lishi
ng
Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany bull Im
age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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pan
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
copy H
oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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oug
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
copy H
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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A B
C
GH
E F
D
P
81 mm
152 mm
125 mm17 in
6 in4 in
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Compare the volume of a square pyramid to the volume of a square prism with the same base and height as the pyramid
2 Which of the following equations could describe a square pyramid Select all that apply
A 3Vh = B
B V = 1 _ 3 ℓwB
C w = 3V _ ℓh
D V _ B = h _ 3
E V = w 2 h _ 3
F 1 _ 3 = VBh
3 Justify Reasoning As shown in the figure polyhedron ABCDEFGH is a cube and P is any point on face EFGH Compare the volume of the pyramid PABCD and the volume of the cube Demonstrate how you came to your answer
Find the volume of the pyramid Round your answer to the nearest tenth
4 5
Module 21 1139 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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oug
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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oug
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Mif
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cour
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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arcourt Publishin
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pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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pan
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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arcourt Publishin
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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pan
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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pany
Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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copy H
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arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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oug
hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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hton Mifflin H
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm
12 cm
12 cm
18 cm
125 cm
75cm
25 cm
5 cm
4 cm
4radic3 cm
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pany
6 Find the volume of a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft
7 The area of the base of a hexagonal pyramid is 24 ______ tan 30deg cm 2 Find its volume
Find the volume of the composite figure Round to the nearest tenth
8 9
10 Given a square pyramid with a height of 21 ft and a volume of 3969 cubic feet find the length of one side of the square base Round to the nearest tenth
11 Consider a pyramid with height 10 feet and a square base with side length of 7 feet How does the volume of the pyramid change if the base stays the same and the height is doubled
Module 21 1140 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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cour
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Com
pan
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
4 ft
7 ft7 ft
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pany
16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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cour
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pan
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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pan
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Mif
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cour
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Com
pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
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18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
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19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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otog
raphyIm
agin
giSto
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12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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hton Mifflin H
arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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10 cm
X
10 cm
15 m
1 m
15 cm
15 cm
25 cm
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12 Algebra Find the value of x if the volume of the pyramid shown is 200 cubic centimeters
13 Find the height of a rectangular pyramid with length 3 meters width 8 meters and volume 112 cubic meters
14 A storage container for grain is in the shape of a square pyramid with the dimensions shown
a What is the volume of the container in cubic centimeters
b Grain leaks from the container at a rate of 4 cubic centimeters per second Assuming the container starts completely full about how many hours does it take until the container is empty
15 A piece of pure copper in the shape of a rectangular pyramid with the dimensions shown has a mass of 1676 grams What is the density of copper Round to the nearest hundredth (Hint density = mass _
volume )
Module 21 1141 Lesson 2
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4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
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10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
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Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
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19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
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g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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4 ft
7 ft7 ft
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16 Represent Real World Problems An art gallery is a 6 story square pyramid with base area 1 __ 2 acre (1 acre = 4840 yd 2 1 story asymp 10 ft) Estimate the volume in cubic yards and cubic feet
17 Analyze Relationships How would the volume of the pyramid shown change if each dimension were multiplied by 6 Explain how you found your answer
18 Geology A crystal is cut into a shape formed by two square pyramids joined at the base Each pyramid has a base edge length of 57 mm and a height of 3 mm What is the volume of the crystal to the nearest cubic millimeter
19 A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards What is the volume of the attic in cubic feet In cubic yards
ge07sec10l07003a AB
3 mm
57 mm45 ft
ge07se_c10l07004a
5 yd
Module 21 1142 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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ina81iSto
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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gie
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asse
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hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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hton Mifflin H
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
10 ft
V = (49)(10)12
= 245 ft3
7 ft
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HOT Focus on Higher Order Thinking
20 Explain the Error Describe and correct the error in finding the volume of the pyramid
21 Communicate Mathematical Ideas A pyramid has a square base and a height of 5 ft The volume of the pyramid is 60 ft 3 Explain how to find the length of a side of the pyramidrsquos base
22 Critical Thinking A rectangular pyramid has a base length of 2 a base width of x and a height of 3x Its volume is 512 cm 3 What is the area of the base
Module 21 1143 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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pany
Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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Com
pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
Genna is making a puzzle using a wooden cube Shersquos going to cut the cube into three pieces The figure below shows the lines along which she plans to cut away the first piece The result will be a piece with four triangular sides and a square side (shaded)
1 Each cut Genna makes will begin at the upper left corner of the cube Write a rule describing where she drew the lines for the first piece
2 The figure below shows two of the lines along which Genna will cut the second piece Draw a cube and on it draw the two lines Genna drew Then using the same rule you used above draw the third line and shade the square base of the second piece
3 When Genna cut away the second piece of the puzzle the third piece remained Draw a new cube and then draw the lines that mark the edges of the third piece Shade the square bottom of the third piece
4 Compare the volumes of the three pieces Explain your reasoning
5 Explain how the model confirms the formula for the volume of a pyramid
Module 21 1144 Lesson 2
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
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19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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otog
raphyIm
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giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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P Im
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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Base of inscribedpyramid has 3 sides
Base of inscribedpyramid has 4 sides
Base of inscribedpyramid has 5 sides
O
rry
12
MxA B
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Name Class Date
Explore Developing a Volume FormulaYou can approximate the volume of a cone by finding the volumes of inscribed pyramids
A The base of a pyramid is inscribed in the circular base of the cone and is a regular n-gon Let O be the center of the conersquos base let r be the radius of the cone and let h be the height of the cone Draw radii from O to the vertices of the n-gon
Construct segment _ OM from O to the midpoint M of
_ AB How can you prove that
AOM cong BOM
B How is ang1 cong ang2
Resource Locker
Module 21 1145 Lesson 3
213 Volume of ConesEssential Question How do you calculate the volumes of composite figures that include cones
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C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
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18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
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19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
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43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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redits
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
C How many triangles congruent to AOB surround point O to make up the n-gon that is the base of the pyramid How can this be used to find the angle measures of AOM and BOM
D In AOM sin ang1 = x _ r so x = rsin ang1 In AOM cos ang1 = y _ r so y = rcos ang1
Since ang1 has a known value rewrite x and y using substitution
E To write an expression for the area of the base of the pyramid first write an expression for the area of AOB
Area of AOB = 1 _ 2 sdot base sdot height
= 1 _ 2 sdot 2x sdot y
= xy
What is the area of AOB substituting the new values for x and y What is the area of the n triangles that make up the base of the pyramid
F Use the area of the base of the pyramid to find an equation for the volume of the pyramid
Module 21 1146 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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cour
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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arcourt Publishin
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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arcourt Publishin
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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cour
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pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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oug
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cour
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lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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age C
redits
copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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oug
hton Mifflin H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Your expression for the pyramidrsquos volume includes the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) Use a calculator as follows to discover what happens to this expression as n gets larger and larger
bull Enter the expression n sin ( 180deg _ n ) cos ( 180deg _ n ) as Y 1 using x for n
bull Go to the Table Setup menu and enter the values shown
bull View a table for the function and scroll down
What happens to the expression as n gets very large
If n sin ( 180deg _ n ) cos ( 180deg _ n ) gets closer to π as n becomes greater what happens to the entire expression for the volume of the inscribed pyramid How is the area of the circle related to the expression for the base
Reflect
1 How is the formula for the volume of a cone related to the formula for the volume of a pyramid
Explain 1 Finding the Volume of a ConeThe volume relationship for cones that you found in the Explore can be stated as the following formula
Volume of a Cone
The volume of a cone with base radius r and base area B = π r 2 and height h is given by V = 1 __ 3 Bh or by V = 1 __ 3 π r 2 h
You can use a formula for the volume of a cone to solve problems involving volume and capacity
Module 21 1147 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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arcourt Publishin
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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oug
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
18 in
22 in
39 in
24 in
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pany
Example 1 The figure represents a conical paper cup How many fluid ounces of liquid can the cup hold Round to the nearest tenth (Hint 1 in 3 asymp 0554 fl oz)
A Find the radius and height of the cone to the nearest hundredth
The radius is half of the diameter so r = 1 _ 2 (22 in) = 11 in
To find the height of the cone use the Pythagorean Theorem
r 2 + h 2 = (18) 2
(11) 2 + h 2 = (18) 2
121 + h 2 = 324
h 2 = 203 so h asymp 142 in
B Find the volume of the cone in cubic inches
V = 1 _ 3 π r 2 h asymp 1 _ 3 π ( ) 2 ( ) asymp i n 3
C Find the capacity of the cone to the nearest tenth of a fluid ounce
i n 3 asymp i n 3 times 0554 fl oz _ 1 i n 3
asymp fl oz
Your Turn
Right after Cindy buys a frozen yogurt cone her friend Maria calls her and they talk for so long that the frozen yogurt melts before Cindy can eat it The cone has a slant height of 39 in and a diameter of 24 in If the frozen yogurt has the same volume before and after melting and when melted just fills the cone how much frozen yogurt did Cindy have before she talked to Maria to the nearest tenth of a fluid ounce
2 Find the radius Then use the Pythagorean Theorem to find the height of the cone
3 Find the volume of the cone in cubic inches
4 Find the capacity of the cone to the nearest fluid ounce
Module 21 1148 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Com
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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arcourt Publishin
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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Mif
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cour
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lishi
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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arcourt Publishin
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
copy H
oug
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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cour
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lishi
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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ina81iSto
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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mag
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm32 mm
16 mm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic millimeter
A Find the volume of the cylinder
First find the radius r = 1 _ 2 (16 mm) = 8 mm
V = π r 2 h = π (8) 2 (19) = 3820176 hellip m m 3
B Find the volume of the cone
The height of the cone is h = mm - mm = mm
It has the same radius as the cylinder r = mm
V = 1 _ 3 π r 2 h = 1 _ 3 π ( ) 2
( ) asymp m m 3
C Find the total volume
Total volume = volume of cylinder + volume of cone
= m m 3 + m m 3
asymp m m 3
Reflect
5 Discussion A composite figure is formed from a cone and a cylinder with the same base radius and its volume can be calculated by multiplying the volume of the cylinder by a rational number a _
b What
arrangements of the cylinder and cone could explain this
Module 21 1149 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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cour
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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oug
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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arcourt Publishin
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pany bull
copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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pan
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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pan
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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gie
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asse
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hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
43 cm
36 cm
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pany
Your Turn
Making a cone-shaped hole in the top of a cylinder forms a composite figure so that the apex of the cone is at the base of the cylinder Find the volume of the figure to the nearest tenth
6 Find the volume of the cylinder
7 Find the volume of the figure
Elaborate
8 Could you use a circumscribed regular n-gon as the base of a pyramid to derive the formula for the volume of a cone Explain
9 Essential Question Check-In How do you calculate the volumes of composite figures that include cones
Module 21 1150 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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cour
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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mag
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red
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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19 mm
42 mm
59 ft
63 ft
20 cm
22 cm
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
1 Interpret the Answer Katherine is using a cone to fill a cylinder with sand If the radii and height are equal on both objects and Katherine fills the cone to the very top how many cones will it take to fill the cylinder with sand Explain your answer
Find the volume of the cone Round the answer to the nearest tenth
2 3
4
Module 21 1151 Lesson 3
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30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
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13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Com
pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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pany
Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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mag
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
30 in
24 in
41 m
9 m
6 in
4 in 8 in
12 in
6 ft
10 ft
copy H
oug
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arcourt Publishin
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pany
Find the volume of the cone Leave the answer in terms of π
5 6
Find the volume of the composite figures Round the answer to the nearest tenth
7 8
Module 21 1152 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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Com
pan
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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arcourt Publishin
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
x
x
x
8 in
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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cour
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Com
pan
y
Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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pany
C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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cour
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lishi
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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arcourt Publishin
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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oug
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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Mif
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cour
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lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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redits
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
oug
hton Mifflin H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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oug
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
13 m
2 m
1 m 12 ft
3 ft
5 ft10 ft
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9 10
11 Match the dimensions of a cone on the left with its volume on the right
A radius 3 units height 7 units 25π ___ 6 units 3
B diameter 5 units height 2 units 240π units 3
C radius 28 units slant height 53 units 11760π units 3
D diameter 24 units slant height 13 units 21π units 3
Module 21 1153 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJenniferPh
otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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oug
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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oug
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lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
e C
red
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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hton Mifflin H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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ough
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Miff
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hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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hton Mifflin H
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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otog
raphyIm
agin
giSto
ckPhotocom
12 The roof of a grain silo is in the shape of a cone The inside radius is 20 feet and the roof is 10 feet tall Below the cone is a cylinder 30 feet tall with the same radius
a What is the volume of the silo
b If one cubic foot of wheat is approximately 48 pounds and the farmerrsquos crop consists of approximately 2 million pounds of wheat will all of the wheat fit in the silo
13 A cone has a volume of 18π in 3 Which are possible dimensions of the cone Select all that apply
A diameter 1 in height 18 in
B diameter 6 in height 6 in
C diameter 3 in height 6 in
D diameter 6 in height 3 in
E diameter 4 in height 135 in
F diameter 135 in height 4 in
Module 21 1154 Lesson 3
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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oug
hton Mifflin H
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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cour
t Pub
lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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oug
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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oug
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
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Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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6 ft
8 ft
10 ft 12 in
12 in
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14 The figure shows a water tank that consists of a cylinder and a cone How many gallons of water does the tank hold Round to the nearest gallon (Hint 1 ft 3 = 748 gal)
15 Roland is using a special machine to cut cones out of cylindrical pieces of wood The machine is set to cut out two congruent cones from each piece of wood leaving no gap in between the vertices of the cones What is the volume of material left over after two cones are cut out
Module 21 1155 Lesson 3
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x
x
x
8 in
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pany
16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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pan
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
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arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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16 Algebra Develop an expression that could be used to solve for the volume of this solid for any value of x
17 Persevere in Problem Solving A juice stand sells smoothies in cone-shaped cups that are 8 in tall The regular size has a 4 in diameter The jumbo size has an 8 in diameter
a Find the volume of the regular size to the nearest tenth
b Find the volume of the jumbo size to the nearest tenth
c The regular size costs $125 What would be a reasonable price for the jumbo size Explain your reasoning
18 Find the volume of a cone with base area 36π ft 2 and a height equal to twice the radius
19 Find the base circumference of a cone with height 5 cm and volume 125π cm 3
Module 21 1156 Lesson 3
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12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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20 in12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
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Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 cm12 cm
20 cm
= _______1088π3
cm3
( 82π)( 17)__13=V V
= cm3
( 82π)( 15)__13=
320π8 cm
17 cm
15 cmA B
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oug
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Mif
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Har
cour
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HOT Focus on Higher Order Thinking
20 Analyze Relationships Popcorn is available in two cups a square pyramid or a cone as shown The price of each cup of popcorn is the same Which cup is the better deal Explain
21 Make a Conjecture A cylinder has a radius of 5 in and a height of 3 in Without calculating the volumes find the height of a cone with the same base and the same volume as the cylinder Explain your reasoning
22 Analyze Relationships A sculptor removes a cone from a cylindrical block of wood so that the vertex of the cone is the center of the cylinderrsquos base as shown Explain how the volume of the remaining solid compares with the volume of the original cylindrical block of wood
23 Explain the Error Which volume is incorrect Explain the error
Module 21 1157 Lesson 3
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Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
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r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
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Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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y
Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Zone 1
Zone 2
30deg30deg
3 m3 m
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Lesson Performance Task
Yoursquove just set up your tent on the first night of a camping trip that yoursquove been looking forward to for a long time Unfortunately mosquitoes have been looking forward to your arrival even more than you have When you turn on your flashlight you see swarms of themmdashan average of 800 mosquitoes per square meter in fact
Since yoursquore always looking for a way to use geometry you decide to solve a problem How many mosquitoes are in the first three meters of the cone of your flashlight (Zone 1 in the diagram) and how many are in the second three meters (Zone 2)
1 Explain how you can find the volume of the Zone 1 cone
2 Find the volume of the Zone 1 cone Write your answer in terms of π
3 Explain how you can find the volume of the Zone 2 cone
4 Find the volume of the Zone 2 cone Write your answer in terms of π
5 How many more mosquitoes are there in Zone 2 than there are in Zone 1 Use 314 for π
Module 21 1158 Lesson 3
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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pan
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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atur
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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hton Mifflin H
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r
r r
r
r
x
R
x
xr
r
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Name Class Date
Explore Developing a Volume FormulaTo find the volume of a sphere compare one of its hemispheres to a cylinder of the same height and radius from which a cone has been removed
A The region of a plane that intersects a solid figure is called a cross section To show that cross sections have the same area at every level use the Pythagorean Theorem to find a relationship between r x and R
B A cross section of the cylinder with the cone removed is a ring
To find the area of the ring find the area of the outer circle and of the inner circle Then subtract the area of the inner circle from the outer circle
Resource Locker
Module 21 1159 Lesson 4
214 Volume of SpheresEssential Question How can you use the formula for the volume of a sphere to calculate the
volumes of composite figures
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
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cour
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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hton Mifflin H
arcourt Publishin
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
arcourt Publishin
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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oug
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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C Find an expression for the volume of the cylinder with the cone removed
D Use Cavalierirsquos principle to deduce the volume of a sphere with radius r
Reflect
1 How do you know that the height h of the cylinder with the cone removed is equal to the radius r
2 What happens to the cross-sectional areas when x = 0 when x = r
Module 21 1160 Lesson 4
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72 ft
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
72 ft
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pan
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Explain 1 Finding the Volume of a SphereThe relationship you discovered in the Explore can be stated as a volume formula
Volume of a Sphere
The volume of a sphere with radius r is given by V = 4 _ 3 π r 3
You can use a formula for the volume of a sphere to solve problems involving volume and capacity
Example 1 The figure represents a spherical helium-filled balloon This tourist attraction allows up to 28 passengers at a time to ride in a gondola suspended underneath the balloon as it cruises at an altitude of 500 ft How much helium to the nearest hundred gallons does the balloon hold Round to the nearest tenth (Hint 1 gal asymp 01337 ft 3 )
Step 1 Find the radius of the balloon
The radius is half of the diameter so r = 1 _ 2 (72 ft) = 36 ft
Step 2 Find the volume of the balloon in cubic feet
V = 4 _ 3 π r 3
= 4 _ 3 π ( ) 3
asymp ft 3
Step 3 Find the capacity of the balloon to the nearest gallon
ft 3 asymp ft 3 times 1 gal
_ 01337 ft 3
asymp gal
Your Turn
A spherical water tank has a diameter of 27 m How much water can the tank hold to the nearest liter (Hint 1000 L = 1 m3)
3 Find the volume of the tank in cubic meters
4 Find the capacity of the tank to the nearest liter
Module 21 1161 Lesson 4
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7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
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gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
7 in
13 cm5 cm
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Explain 2 Finding the Volume of a Composite FigureYou can find the volume of a composite figure using appropriate volume formulas for the different parts of the figure
Example 2 Find the volume of the composite figure Round to the nearest cubic centimeter
Step 1 Find the volume of the hemisphere
Step 2 Find the height of the cone
h 2 + ( ) 2
= ( ) 2
h 2 + =
h 2 =
h =
Step 3 Find the volume of the cone
The cone has the same radius as the
hemisphere r = cm
V = 1 _ 3 π r 2 h
= 1 _ 3 π ( ) 2 ( ) = c m 3
Step 4 Find the total volume
Total volume = volume of hemisphere + volume of cone
= c m 3 + c m 3
asymp cm 3
Reflect
5 Is it possible to create a figure by taking a cone and removing from it a hemisphere with the same radius
Your Turn
6 A composite figure is a cylinder with a hemispherical hole in the top The bottom of the hemisphere is tangent to the base of the cylinder Find the volume of the figure to the nearest tenth
Module 21 1162 Lesson 4
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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cour
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lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mag
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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pan
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
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arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
arcourt Publishin
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
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Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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Elaborate
7 Discussion Could you use an inscribed prism to derive the volume of a hemisphere Why or why not Are there any other ways you could approximate a hemisphere and what problems would you encounter in finding its volume
8 Essential Question Check-In A gumball is in the shape of a sphere with a spherical hole in the center How might you calculate the volume of the gumball What measurements are needed
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Analyze Relationships Use the diagram of a sphere inscribed in a cylinder to describe the relationship between the volume of a sphere and the volume of a cylinder
Evaluate Homework and Practice
Module 21 1163 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
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ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
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Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Circumferenceof great circleis 14π cm
37 in11 ft
20 cm 1 m circle is 81π in2Area of great
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pany
Find the volume of the sphere Round the answer to the nearest tenth
2 3 4
Find the volume of the sphere Leave the answer in terms of π
5 6 7
Module 21 1164 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
copy H
oug
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Mif
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cour
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lishi
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Com
pan
yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pany
For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Com
pan
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
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g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
copy H
oug
hton Mifflin H
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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oug
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
copy H
oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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cour
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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2 ft5 ft 8 in
3 in
2 in
10 cm8 cm
3 cm
4 cm
24 mm
8 mm
10 mm
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yFind the volume of the composite figure Leave the answer in terms of π
8 9
Find the volume of the composite figure Round the answer to the nearest tenth
10 11
Module 21 1165 Lesson 4
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ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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copyJasm
ina81iSto
ckPhotocom
12 Analyze Relationships Approximately how many times as great is the volume of a grapefruit with diameter 10 cm as the volume of a lime with diameter 5 cm
13 A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter Estimate the volume of the bead to the nearest whole
14 Algebra Write an expression representing the volume of the composite figure formed by a hemisphere with radius r and a cube with side length 2r
Module 21 1166 Lesson 4
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
5 ft
20 in12 in
12 in
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cour
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pan
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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Miff
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urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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5 ft
20 in12 in
12 in
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15 One gallon of propane yields approximately 91500 BTU About how many BTUs does the spherical storage tank shown provide Round to the nearest million BTUs (Hint 1 f t 3 asymp 748 gal)
16 The aquarium shown is a rectangular prism that is filled with water You drop a spherical ball with a diameter of 6 inches into the aquarium The ball sinks causing the water to spill from the tank How much water is left in the tank Express your answer to the nearest tenth (Hint 1 in 3 asymp 000433 gal)
17 A sphere with diameter 8 cm is inscribed in a cube Find the ratio of the volume of the cube to the volume of the sphere
A 6 _ π
B 2 _ 3π
C 3π _ 4
D 3π _ 2
Module 21 1167 Lesson 4
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
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asse
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hutt
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ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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For Exercises 18ndash20 use the table Round each volume to the nearest billion π
Planet Diameter (mi) Mercury 3032
Venus 7521
Earth 7926
Mars 4222
Jupiter 88846
Saturn 74898
Uranus 31763
Neptune 30775
18 Explain the Error Margaret used the mathematics shown to find the volume of Saturn
V = 4 _ 3 π r 2 = 4 _ 3 π (74898) 2 asymp 4 _ 3 π (6000000000) asymp 8000000000π
Explain the two errors Margaret made then give the correct answer
19 The sum of the volumes of Venus and Mars is about equal to the volume of which planet
20 How many times as great as the volume of the smallest planet is the volume of the largest planet Round to the nearest thousand
Module 21 1168 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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Mif
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cour
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Com
pan
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mag
e C
red
its
copyRe
gie
n Pa
asse
nS
hutt
erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
copy H
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hton Mifflin H
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pany bull Im
age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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hton Mifflin H
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
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Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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asse
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erst
ock
HOT Focus on Higher Order Thinking
21 Make a Conjecture The bathysphere was an early version of a submarine invented in the 1930s The inside diameter of the bathysphere was 54 inches and the steel used to make the sphere was 15 inches thick It had three 8-inch diameter windows Estimate the volume of steel used to make the bathysphere
22 Explain the Error A student solved the problem shown Explain the studentrsquos error and give the correct answer to the problem
A spherical gasoline tank has a radius of 05 ft When filled the tank provides 446483 BTU How many BTUs does one gallon of gasoline yield Round to the nearest thousand BTUs and use the fact that 1 ft 3 asymp 748 gal
23 Persevere in Problem Solving The top of a gumball machine is an 18 in sphere The machine holds a maximum of 3300 gumballs which leaves about 43 of the space in the machine empty Estimate the diameter of each gumball
The volume of the tank is 4 __ 3 π r 3 = 4 __ 3 π (05) 3 ft 3 Multiplying by 748 shows that this is approximately 392 gal So the number of BTUs in one gallon of gasoline is approximately 446483 times 392 asymp 1750000 BTU
Module 21 1169 Lesson 4
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Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
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IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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atur
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oughton Mifflin H
arcourt Publishing Company
Lesson Performance Task
For his science project Bizbo has decided to build a scale model of the solar system He starts with a grapefruit with a radius of 2 inches to represent Earth His ldquoEarthrdquo weighs 05 pounds
Find each of the following for Bizborsquos model Use the rounded figures in the table Round your answers to two significant figures Use 314 for π
1 the scale of Bizborsquos model 1 inch = miles
2 Earthrsquos distance from the Sun in inches and in miles
3 Neptunersquos distance from the Sun in inches and in miles
4 the Sunrsquos volume in cubic inches and cubic feet
5 the Sunrsquos weight in pounds and in tons (Note the Sunrsquos density is 026 times the Earthrsquos density)
Radius (mi) Distance from Sun (mi)
Earth 4 times 1 0 3 93 times 1 0 7
Neptune 15 times 1 0 4 28 times 1 0 9
Sun 43 times 1 0 5
Module 21 1170 Lesson 4
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L4indd 1170 42717 417 PM
0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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pan
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mag
e C
red
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copyRe
x Fe
atur
esA
P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
6 ft
5 ft
12 in
3 in
8 in
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pany
Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
copy H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
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0-2-3
-3
-2
1
2
3y
1 2 3
x
A (-2 -1) D (1 -1)
B (0 1) C (3 1)
Name Class Date
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atur
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P Im
ages
Explore Exploring Effects of Changing Dimensions on Perimeter and Area
Changes made to the dimensions of a figure can affect the perimeter and the area
Use the figure to investigate how changing one or more dimensions of the figure affect its perimeter and area
A Apply the transformation (x y) rarr (3x y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x y)
P = 6 + 4 radic_
2 P =
A = 6 A =
B Apply the transformation (x y) rarr (x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
C Apply the transformation (x y) rarr (3x 3y) Find the perimeter and the area
Original Dimensions Dimensions after (x y) rarr (3x 3y)
P = 6 + 4 radic_
2 P =
A = 6 A =
215 Scale FactorEssential Question How does multiplying one or more of the dimensions of a figure affect its
attributes
Resource Locker
Resource Locker
Module 21 1171 Lesson 5
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6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
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10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
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12
12 ft
6 ft
3 in
4 in
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age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
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12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
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768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
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95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
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Module 21 1182 Study Guide Review
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Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
6 ft
5 ft
12 in
3 in
8 in
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Reflect
1 Describe the changes that occurred in Steps A and B Did the perimeter or area change by a constant factor
2 Describe the changes that occurred in Step C Did the perimeter or area change by a constant factor
Explain 1 Describe a Non-Proportional Dimension ChangeIn a non-proportional dimension change you do not use the same factor to change each dimension of a figure
Example 1 Find the area of the figure
A Find the area of the parallelogram Then multiply the length by 2 and determine the new area Describe the changes that took place
B Find the area of the trapezoid Then multiply the height by 05 and determine the new area Describe the changes that took place
Original Figure A = 1 _ 2 ( b 1 + b 2 ) h =
Transformed Figure A = 1 _ 2 ( b 1 + b 2 ) h =
When the height of the trapezoid changes by a factor of the
area of the trapezoid changes by a factor of
Reflect
3 Discussion When a non-proportional change is applied to the dimensions of a figure does the perimeter change in a predictable way
Your Turn
4 Find the area of a triangle with vertices (-5 -2) (-5 7) and (3 1) Then apply the transformation (x y) rarr (x 4y) and determine the new area Describe the changes that took place
Original Figure Transformed Figure
A = bh = 6 sdot 5 = 30 ft 2 A = bh = 12 sdot 5 = 60 ft 2
When the length of the parallelogram changes by a factor of 2 the area changes by a factor of 2
Module 21 1172 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
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height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
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3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
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h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
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yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
10 cm 4 cm
4
6
6
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5 Find the area of the figure Then multiply the width by 5 and determine the new area Describe the changes that took place
Explain 2 Describe a Proportional Dimension ChangeIn a proportional dimension change you use the same factor to change each dimension of a figure
Example 2 Find the area and perimeter of a circle
A Find the circumference and area of the circle Then multiply the radius by 3 and find the new circumference and area Describe the changes that took place
Original Figure C = 2π (4) = 8π
A = π (4) 2 = 16π
Transformed Figure C = 2π (12) = 24π
A = π (12) 2 = 144π
The circumference changes by a factor of 3 and the area changes by a factor of 9 or 3 2
B Find the perimeter and area of the figure Then multiply the length and height by 1 __ 3 and find the new perimeter and area Describe the changes that took place
Original Figure Transformed Figure
P = P =
A = A =
The perimeter changes by a factor of and the area changes by a factor of
Reflect
6 Fill in the table to describe the effect on perimeter (or circumference) and area when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Perimeter or Circumference Area
All dimensions multiplied by a
Module 21 1173 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12
12 ft
6 ft
3 in
4 in
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Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
3 in
3 in
8 in
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pan
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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hton Mifflin H
arcourt Publishin
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12
12 ft
6 ft
3 in
4 in
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age C
redits copy
Ocean
C
orbis
Your Turn
7 Find the circumference and area of the circle Then multiply the radius by 025 and find the new circumference and area Describe the changes that took place
Explain 3 Describe a Proportional Dimension Change for a Solid
In a proportional dimension change to a solid you use the same factor to change each dimension of a figure
Example 3 Find the volume of the composite solid
A A company is planning to create a similar version of this storage tank a cylinder with hemispherical caps at each end Find the volume and surface area of the original tank Then multiply all the dimensions by 2 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = π r 2 h + 4 __ 3 π r 3 and the surface area is S = 2πrh + 4π r 2
Original Solid Transformed Solid
V = π (3) 2 (12) + 4 _ 3 π (3) 3 = 144π cu ft V = π (6) 2 (24) + 4 _ 3 π (6) 3 = 1152π cu ft
S = 2π (3 sdot 12) + 4π (3) 2 = 108π sq ft S = 2π (6 sdot 24) + 4π (6) 2 = 432π sq ft
The volume changes by a factor of 8 and the surface area changes by a factor of 4
B A childrenrsquos toy is shaped like a hemisphere with a conical top A company decides to create a smaller version of the toy Find the volume and surface area of the original toy Then multiply all dimensions by 2 __ 3 and find the new volume and surface area Describe the changes that took place
The volume of the solid is V = 1 __ 3 π r 2 h + 2 __ 3 π r 3
and the surface area is S = πr radic_
r 2 + h 2 + 2π r 2
Original Solid Transformed Solid
V = cu in V = cu in
S = sq in S = sq in
The volume changes by a factor of and the surface area changes by a factor of
Module 21 1174 Lesson 5
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3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
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Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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hton Mifflin H
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
3 in
3 in
8 in
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Reflect
8 Fill in the table to describe the effect on surface area and volume when the dimensions of a figure are changed proportionally
Effects of Changing Dimensions Proportionally
Change in Dimensions Surface Area Volume
All dimensions multiplied by a
Your Turn
9 A farmer has made a scale model of a new grain silo Find the volume and surface area of the model Use the scale ratio 1 36 to find the volume and surface area of the silo Compare the volumes and surface areas relative to the scale ratio Be consistent with units of measurement
Elaborate
10 Two square pyramids are similar If the ratio of a pair of corresponding edges is a b what is the ratio of their volumes What is the ratio of their surface areas
11 Essential Question Check-In How is a non-proportional dimension change different from a proportional dimension change
Module 21 1175 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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pan
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
arcourt Publishin
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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hton Mifflin H
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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cour
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
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h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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ough
ton
Miff
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urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
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pany
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
A trapezoid has the vertices (0 0) (4 0) (4 4) and (-3 4)
1 Describe the effect on the area if only the x-coordinates of the vertices are multiplied by 1 __ 2
2 Describe the effect on the area if only the y-coordinates of the vertices are multiplied by 1 __ 2
3 Describe the effect on the area if both the x- and y-coordinates of the vertices are multiplied by 1 __ 2
4 Describe the effect on the area if the x-coordinates are multiplied by 2 and y-coordinates are multiplied by 1 __ 2
Module 21 1176 Lesson 5
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-BCA-B
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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oug
hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
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11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
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IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
12 m
21 m
24 in
9 in
6 ft
18 ft
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Describe the effect of the change on the area of the given figure
5 The height of the triangle is doubled 6 The height of a trapezoid with base lengths 12 cm and 8 cm and height 5 cm is multiplied by 1 __ 3
7 The base of the parallelogram is multiplied by 2 __ 3 8 Communicate Mathematical Ideas A triangle has vertices (1 5) (2 3) and (-1 -6) Find the effect that multiplying the height of the triangle by 4 has on the area of the triangle without doing any calculations Explain
Describe the effect of each change on the perimeter or circumference and the area of the given figure
9 The base and height of an isosceles triangle with base 12 in and height 6 in are both tripled
10 The base and height of the rectangle are both multiplied by 1 __ 2
Module 21 1177 Lesson 5
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2 yd
3 yd10 m
5 m
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hton Mifflin H
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 yd
3 yd10 m
5 m
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pany
11 The dimensions are multiplied by 5 12 The dimensions are multiplied by 3 _ 5
13 For each change check whether the change is non-proportional or proportionalA The height of a triangle is doubled proportional non-proportional
B All sides of a square are quadrupled proportional non-proportional
C The length of a rectangle is multiplied by 3 _ 4 proportional non-proportional
D The height of a triangular prism is tripled proportional non-proportional
E The radius of a sphere is multiplied by radic ― 5 proportional non-proportional
14 Tina and Kleu built rectangular play areas for their dogs The play area for Tinarsquos dog is 15 times as long and 15 times as wide as the play area for Kleursquos dog If the play area for Kleursquos dog is 60 square feet how big is the play area for Tinarsquos dog
15 A map has the scale 1 inch = 10 miles On the map the area of Big Bend National Park in Texas is about 125 square inches Estimate the actual area of the park in acres (Hint 1 square mile = 640 acres)
16 A restaurant has a weekly ad in a local newspaper that is 2 inches wide and 4 inches high and costs $3675 per week The cost of each ad is based on its area If the owner of the restaurant decided to double the width and height of the ad how much will the new ad cost
17 Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled Find the perimeter of the new triangle in inches
A rectangular prism has vertices (0 0 0) (0 3 0) (7 0 0) (7 3 0) (0 0 6) (0 3 6) (7 0 6) and (7 3 6)
18 Suppose all the dimensions are tripled Find the new vertices
19 Find the effect of the change on the volume of the prism
Module 21 1178 Lesson 5
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
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oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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y
Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
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84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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hton Mifflin H
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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20 How would the effect of the change be different if only the height had been tripled
21 Analyze Relationships How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram if the parallelogram does have to be similar to the original parallelogram
HOT Focus on Higher Order Thinking
22 Algebra A square has a side length of (2x + 5) cm
a If the side length is mulitplied by 5 what is the area of the new square
b Use your answer to part (a) to find the area of the original square without using the area formula Justify your answer
23 Algebra A circle has a diameter of 6 in If the circumference is multiplied by (x + 3) what is the area of the new circle Justify your answer
24 Communicate Mathematical Ideas The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a similar prism Make a conjecture about the ratio of the surface area of the new prism to its volume Test your conjecture using a cube with an edge length of 1 and a scale factor of 2
Module 21 1179 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
arcourt Publishin
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
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14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
768 pixels
1024 pixels
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pany
On a computer screen lengths and widths are measured not in inches or millimeters but in pixels A pixel is the smallest visual element that a computer is capable of processing A common size for a large computer screen is 1024 times 768 pixels (Widths rather than heights are conventionally listed first) For the following assume yoursquore working on a 1024 times 768 screen
1 You have a photo measuring 640 times 300 pixels and you want to enlarge it proportionally so that it is as wide as the computer screen Find the measurements of the photo after it has been scaled up Explain how you found the answer
2 a Explain why you canrsquot enlarge the photo proportionally so that it is as tall as the computer screen
b Why canrsquot you correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 divide 640 and the height by a factor of 768 divide 300
3 You have some square photos and you would like to fill the screen with them so there is no overlap and there are no gaps between photos Find the dimensions of the largest such photos you can use (all of them the same size) and find the number of photos Explain your reasoning
Lesson Performance Task
Module 21 1180 Lesson 5
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-BCA-B
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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cour
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pan
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
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Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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Miff
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ublis
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Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
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10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
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IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
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pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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lishi
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Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
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g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Volume = 1 cubic unit
axis
right cylinderright prism
area is B square units
height is 1 unit
axis
right cylinderright prism
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Name Class Date
Explore Developing a Basic Volume FormulaThe volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure This prism is made up of 8 cubes each with a volume of 1 cubic centimeter so it has a volume of 8 cubic centimeters You can use this idea to develop volume formulas
In this activity yoursquoll explore how to develop a volume formula for a right prism and a right cylinder
A right prism has lateral edges that are perpendicular to the bases with faces that are all rectangles
A right cylinder has bases that are perpendicular to its center axis
A On a sheet of paper draw a quadrilateral shape Make sure the sides arenrsquot parallel Assume the figure has an area of B square units
B Use it as the base for a prism Take a block of Styrofoam and cut to the shape of the base Assume the prism has a height of 1 unit
How would changing the area of the base change the volume of the prism
Resource Locker
Module 21 1121 Lesson 1
211 Volume of Prisms and CylindersEssential Question How do the formulas for the volume of a prism and cylinder relate to area
formulas that you already know
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
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pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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pan
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
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Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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oug
hton Mifflin H
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pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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Miff
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Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
hton
Mif
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Har
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t Pub
lishi
ng
Com
pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
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Har
cour
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lishi
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pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
height is h units
B
W
ℓ
h
S
S
Sh B
W
ℓ
h
S
S
Sh
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
If the base has an area of B square units how many cubic units does the prism contain
Now use the base to build a prism with a height of h units
How much greater is the volume of this prism compared to the one with a height of 1
Reflect
1 Suppose the base of the prism was a rectangle of sides l and w Write a formula for the volume of the prism using l w and h
2 A cylinder has a circular base Use the results of the Explore to write a formula for the volume of a cylinder Explain what you did
Explain 1 Finding the Volume of a PrismThe general formula for the volume of a prism is V = B ∙ h With certain prisms the volume formula can include the formula for the area of the base
Volume of a Prism
The formula for the volume of a right rectangular prism with length ℓ width w and height h is V = ℓwh
The formula for the volume of a cube with edge length s is V = s 3
Module 21 1122 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Har
cour
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Com
pan
y
Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
y
Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
arcourt Publishin
g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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ough
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Miff
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urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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Miff
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Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
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Mif
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cour
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
3 cm
120 ft
60 ft
8 ft
ge07se_c10l06003aAB
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cour
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Com
pan
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Example 1 Use volume formulas to solve real world problems
A A shark and ray tank at the aquarium has the dimensions shown Estimate the volume of water in gallons Use the conversion 1 gallon = 0134 f t 3
Step 1 Find the volume of the aquarium in cubic feet
V = ℓwh = (120) (60) (8) = 57600 f t 3
Step 2 Use the conversion factor 1 gallon
_ 0134 f t 3
to estimate
the volume of the aquarium in gallons
57600 f t 3 ∙ 1 gallon
_ 0134 f t 3
asymp 429851 gallons 1 gallon
_ 0134 f t 3
= 1
Step 3 Use the conversion factor 1 gallon
__ 833 pounds
to estimate the weight of the water
429851 gallons ∙ 833 pounds
__ 1 gallon
asymp 3580659 pounds 833 pounds
__ 1 gallon
= 1
The aquarium holds about 429851 in gallons The water in the aquarium weighs about 3580659 pounds
B Chemistry Ice takes up more volume than water This cubic container is filled to the brim with ice Estimate the volume of water once the ice melts
Density of ice 09167 gc m 3 Density of water 1 g cm 3
Step 1 Find the volume of the cube of ice
V = s 3 = = c m 3
Step 2 Convert the volume to mass using the conversion factor
g _
c m 3
c m 3 ∙ g _
c m 3 asymp g
Step 3 Use the mass of ice to find the volume of water Use the conversion factor
248 g ∙ asymp c m 3
Reflect
3 The general formula for the volume of a prism is V = B ∙ h Suppose the base of a prism is a parallelogram of length l and altitude h Use H as the variable to represent the height of the prism Write a volume formula for this prism
Module 21 1123 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Har
cour
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Com
pan
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
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DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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Miff
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urt P
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hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
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hton Mifflin H
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pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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cour
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
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oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
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Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
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arcourt Publishin
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
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Mif
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cour
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lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Volume = 8 cubic unitsEach cube has a
side of 2k
h
h
h
h
h
hh
h
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
Your Turn
4 Find the volume of the figure 5 Find the volume of the figure
Explain 2 Finding the Volume of a CylinderYou can also find the volume of prisms and cylinders whose edges are not perpendicular to the base
Oblique Prism Oblique Cylinder
An oblique prism is a prism that has at least one non-rectangular lateral face
An oblique cylinder is a cylinder whose axis is not perpendicular to the bases
Cavalierirsquos Principle
If two solids have the same height and the same cross-sectional area at every level then the two solids have the same volume
Module 21 1124 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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lishi
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pan
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Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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oug
hton Mifflin H
arcourt Publishin
g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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Miff
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pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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ton
Miff
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pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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hton Mifflin H
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pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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hton
Mif
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cour
t Pub
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pan
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STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
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oug
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cour
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ng
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pan
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bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
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oug
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Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
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hton Mifflin H
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
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Mif
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Har
cour
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lishi
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Com
pan
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
B = 81π cm2
B = 75 cm2
r = 12 inh = 45 in
4x cm5x cm
h = (x + 2) cm
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Com
pan
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Example 2 To find the volume of an oblique cylinder or oblique prism use Cavalierirsquos Principle to find the volume of a comparable right cylinder or prism
A The height of this oblique cylinder is three times that of its radius What is the volume of this cylinder Round to the nearest tenth
Use Cavalierirsquos Principle to find the volume of a comparable right cylinder
Represent the height of the oblique cylinder h = 3r
Use the area of the base to find r π r 2 = 81π c m 2 so r = 9
Calculate the height h = 3r = 27 cm
Calculate the volume V = Bh = (81π) 27 asymp 68707
The volume is about 68707 cubic centimeters
B The height of this oblique square-based prism is four times that of side length of the base What is the volume of this prism Round to the nearest tenth
Calculate the height of the oblique prism
h = s where s is the length of the square base
Use the area of the base to find s
s 2 = c m 2
s = radic_
cm
Calculate the height
h = 4s = 4 cm
Your Turn
Find the volume
6 7
Calculate the volume
V = Bh
= (75 c m 2 ) ( cm) = c m 3
Module 21 1125 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
copy H
oug
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Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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ough
ton
Miff
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hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
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arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
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oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
h = 22 ft
5 ft
B = 36π in2
copy H
oughton Mifflin H
arcourt Publishing Company
Explain 3 Finding the Volume of a Composite FigureRecall that a composite figure is made up of simple shapes that combine to create a more complex shape A composite three-dimensional figure is formed from prisms and cylinders You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure
Example 3 Find the volume of each composite figure
A Find the volume of the composite figure which is an oblique cylinder on a cubic base Round to the nearest tenth
The base area of the cylinder is B = π r 2 = π (5) 2 = 25π f t 2
The cube has side lengths equal to the diameter of the cylinderrsquos circular base s = 10
The height of the cylinder is h = 22 - 10 = 12 ft
The volume of the cube is V = s 3 = 1 0 3 = 1000 f t 3
The volume of the cylinder is V = Bh = (25π f t 2 ) (12 ft) asymp 9425 f t 3
The total volume of the composite figure is the sum of the individual volumes
V = 1000 f t 3 + 9425 f t 3 = 19425 f t 3
B This periscope is made up of two congruent cylinders and two congruent triangular prisms each of which is a cube cut in half along one of its diagonals The height of each cylinder is 6 times the length of the radius Use the measurements provided to estimate the volume of this composite figure Round to the nearest tenth
Use the area of the base to find the radius B = π r 2
π r 2 = π so r = in
Calculate the height each cylinder
h = 6r = 6 ∙ = in
The faces of the triangular prism that intersect the cylinders are congruent squares The side length s of each square is the same as the diameter of the circle
s = d = 2 ∙ = in
The two triangular prisms form a cube What is the volume of this cube
V = s 3 = 3
= i n 3
Find the volume of the two cylinders V = 2 ∙ 36π ∙ = i n 3
The total volume of the composite figure is the sum of the individual volumes
V = i n 3 + i n 3 asymp i n 3
Module 21 1126 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
IN2_MNLESE389847_U9M21L1indd 1126 42717 349 PM
r1 r2
h
r
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hton
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Com
pan
y
Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
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hton Mifflin H
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g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
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ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
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hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
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ough
ton
Miff
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ublis
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Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
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Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
r1 r2
h
r
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Reflect
8 A pipe consists of two concentric cylinders with the inner cylinder hollowed out Describe how you could calculate the volume of the solid pipe Write a formula for the volume
Your Turn
9 This robotic arm is made up of two cylinders with equal volume and two triangular prisms for a hand The volume of each prism is 1 __ 2 r times 1 __ 3 r times 2r where r is the radius of the cylinderrsquos base What fraction of the total volume does the hand take up
Elaborate
10 If an oblique cylinder and a right cylinder have the same height but not the same volume what can you conclude about the cylinders
11 A right square prism and a right cylinder have the same height and volume What can you conclude about the radius of the cylinder and side lengths of the square base
12 Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder
Module 21 1127 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
84 mm35 mm
56 mm
9 yd12 yd
15 yd
4 cm9 cm
6 cm
12 ft
10 ft
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
1 The volume of prisms and cylinders can be represented with Bh where B represents the area of the base Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula
A V = (π r 2 ) h B V = ( 1 _ 2 bh) h C V = ℓwh
Find the volume of each prism or cylinder Round to the nearest hundredth
2 3
4 The area of the hexagonal base is ( 54 ______ tan 30deg ) m 2 Its height is 8 m
5 The area of the pentagonal base is ( 125 _____ tan 36deg ) m 2 Its height is 15 m
6 7
bull Online Homeworkbull Hints and Helpbull Extra Practice
Evaluate Homework and Practice
Module 21 1128 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
14 ft
12 ft
4 ft
4 ft
6 ft 10 in
15 in
5 in
6 cm
4 cm
4 cm4 cm
6 cm
6 cm6 cm 8 cm
8 cm8 cm
2 ft
2 ft
4 ft
4 ft
12 ft
17 cm
14 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
8 Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown What is the volume of the vase in liters Round to the nearest thundredth (Hint Use the right triangle in the cylinder to find its height)
Find the volume of each composite figure Round to the nearest tenth
9 10
11 12 The two figures on each end combine to form a right cylinder
Module 21 1129 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1129 42717 406 PM
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
5 in
1 in
3 in
h
34 cm
60 cm
x
x + 1
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
13 Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle If he wants to fill it to a depth of 4 in how many cubic yards of dirt does he need Round to the nearest cubic yard If dirt costs $25 per y d 3 how much will the project cost
14 Persevere in Problem Solving A cylindrical juice container with a 3 in diameter has a hole for a straw that is 1 in from the side Up to 5 in of a straw can be inserted
a Find the height h of the container to the nearest tenth
b Find the volume of the container to the nearest tenth
c How many ounces of juice does the container hold (Hint 1 i n 3 asymp 055 oz)
15 Abigail has a cylindrical candle mold with the dimensions shown If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm about how many candles can she make after melting the block of wax Round to the nearest tenth
16 Algebra Find the volume of the three-dimensional figure in terms of x
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of
Module 21 1130 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
10 cm 10 cm
10 cmFront Top Side
10 cm
4 cm 4 cm
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
17 One cup is equal to 144375 i n 3 If a 1-cup measuring cylinder has a radius of 2 in what is its height If the radius is 15 in what is its height Round to the nearest tenth
18 Make a Prediction A cake is a cylinder with a diameter of 10 in and a height of 3 in For a party a coin has been mixed into the batter and baked inside the cake The person who gets the piece with the coin wins a prize
a Find the volume of the cake Round to the nearest tenth
b Keka gets a piece of cake that is a right rectangular prism with a 3 in by 1 in base What is the probability that the coin is in her piece Round to the nearest hundredth
HOT Focus on Higher Order Thinking
19 Multi-Step What is the volume of the three-dimensional object with the dimensions shown in the three views
20 Draw Conclusions You can use displacement to find the volume of an irregular object such as a stone Suppose a 2 foot by 1 foot tank is filled with water to a depth of 8 in A stone is placed in the tank so that it is completely covered causing the water level to rise by 2 in Find the volume of the stone
Module 21 1131 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21L1indd 1131 42717 408 PM
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
2 in
11 in
2 in2 in
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
21 Analyze Relationships One juice container is a rectangular prism with a height of 9 in and a 3 in by 3 in square base Another juice container is a cylinder with a radius of 175 in and a height of 9 in Describe the relationship between the two containers
Lesson Performance Task
A full roll of paper towels is a cylinder with a diameter of 6 inches and a hollow inner cylinder with a diameter of 2 inches
1 Find the volume of the paper on the roll Explain your method
2 Each sheet of paper on the roll measures 11 inches by 11 inches by 1 __ 32 inch Find the volume of one sheet Explain how you found the volume
3 How many sheets of paper are on the roll Explain
Module 21 1132 Lesson 1
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
STUDY GUIDE REVIEW MODULE
21Key Vocabularyright prism (prisma recto)
right cylinder (cilindro recto)
oblique prism (prisma oblicuo)
oblique cylinder (cilindro oblicuo)
cross section (seccioacuten transversal)
Write the formula for the volume of a cylinder
Substitute
Simplify
Write the formula for the volume of a pyramid
Substitute
Simplify
Find the radius
Simplify
Write the formula for the volume of a cone
Substitute
Simplify
Essential Question How can you use volume formulasto solve real-world problems
KEY EXAMPLE (Lesson 211)
Find the volume of a cylinder with a base radius of 3 centimeters and a height of 5 centimeters Write an exact answer
V = π r 2 h
= π (3) 2 (5) = 45π c m 3
KEY EXAMPLE (Lesson 212)
Find the volume of a square pyramid with a base side length of12 inches and a height of 7 inches
V = 1 _ 3 Bh
= 1 _ 3 (12) 2 (7)
= 336 in 3
KEY EXAMPLE (Lesson 213)
Find the volume of a cone with a base diameter of 16 feet and aheight of 18 feet Write an exact answer
r = 1 _ 2 (16 ft)
= 8 ft
V = 1 _ 3 π r 2 h
= 1 _ 3 π (8) 2 (18)
= 384π ft 3
KEY EXAMPLE (Lesson 214)
Find the volume of a sphere with a radius of 30 miles Write an exact answer
V = 4 _ 3 π r 3
= 4 _ 3 π (30) 3
= 36000 π mi 3
Volume Formulas
Write the formula for the volume of a sphere
Substitute
Simplify
Module 21 1181 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
95
10
47
63
21
33 cm
16 cm 36 ft
4 ft
5 ft
3 m
8 m 12
EXERCISESFind the volume of each figure Write an exact answer (Lessons 211ndash214)
1 2
3 4
5 6
7 One side of a rhombus measures 12 inches Two angles measure 60deg Find the perimeter and area of the rhombus Then multiply the side lengths by 3 Find the new perimeter and area Describe the changes that took place (Lesson 215)
MODULE PERFORMANCE TASK
How Big Is That SinkholeIn 2010 an enormous sinkhole suddenly appeared in the middle of a Guatemalan neighborhood and swallowed a three-story building above it The sinkhole has an estimated depth of about 100 feet
How much material is needed to fill the sinkhole Determine what information is needed to answer the question Do you think your estimate is more likely to be too high or too low
What are some material options for filling the sinkhole and how much would they cost Which material do you think would be the best choice
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany bull Im
age C
redits copy
String
erReutersC
orbis
Module 21 1182 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
Ready to Go On
44 ft10 ft
c
b
a
lradic2
Top View
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
bull Online Homeworkbull Hints and Helpbull Extra Practice
211ndash215 Volume Formulas
Find the volume of the figure (Lessons 211ndash214)
1 An oblique cylinder next to a cube 2 A prism of volume 3 with a pyramid of the same height cut out
3 A cone with a square pyramid of the same height cut out The pyramid has height l and its square base has area l 2
4 A cube with sides of length s with the biggest sphere that fits in it cut out
ESSENTIAL QUESTION
5 How would you find the volume of an ice-cream cone with ice cream in it What measurements would you need
Module 21 1183 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=NL-BCA-B
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
11 m
3 m
3 2 mradic
10 ft
3 ft2 ft
copy H
oughton Mifflin H
arcourt Publishing Company
Assessment Readiness
Module 21Mixed review
1 A simplified model of a particular monument is a rectangular pyramid placed on top of a rectangular prism as shown The volume of the monument is 66 cubic feet Determine whether the given measurement could be the height of the monumentSelect Yes or No for AndashCA 10 feet Yes NoB 13 feet Yes NoC 15 feet Yes No
2 A standard basketball has a radius of about 47 inches Choose True or False for each statementA The diameter of the basketball is
about 25 inches True FalseB The volume of the basketball is
approximately 2776 i n 3 True FalseC The volume of the basketball is
approximately 4349 i n 3 True False
3 A triangle has a side of length 8 a second side of length 17 and a third side of length x Find the range of possible values for x
4 Find the approximate volume of the figure at right composed of a cone a cylinder and a hemisphere Explain how you found the values needed to compute the volume
Module 21 1184 Study Guide Review
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=NL-ECA-E
IN2_MNLESE389847_U9M21MCindd 1184 42717 428 PM
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
A
C DB
40deg 40deg
A
BC
4y
0 4
x
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
yAssessment readinessUNIT 9 MIXED REVIEW
bull Online Homeworkbull Hints and Helpbull Extra Practice
1 Consider each congruence theorem below Can you use the theorem to determine whether ABC cong ABD
Select Yes or No for AndashC
A ASA Triangle Congruence Theorem Yes No
B SAS Triangle Congruence Theorem Yes No
C SSS Triangle Congruence Theorem Yes No
2 For each pyramid determine whether the statement regarding its volume is true
Select True or False for each statement
A A rectangular pyramid with ℓ = 3 mw = 4 m h = 7 m has volume 84 m 3 True False
B A triangular pyramid with base B = 14 ft 2 and h = 5 ft has volume 60 f t 2 True False
C A pyramid with the same base and heightof a prism has less volume True False
3 For each shape determine whether the statement regarding its volume is true
Select True or False for each statement
A A cone with base radius r = 5 inand h = 12 in has volume 100π i n 3 True False
B A sphere with radius r = 6 _ π m hasvolume 8
_ π 2
m 3 True False
C A sphere is composed of multiplecones with the same radius True False
4 DeMarcus draws ABC Then he translates it along the vector ⟨-4 -3⟩rotates it 180deg and reflects it across the x-axis
Choose True or False for each statement
A The final image of ABC is in Quadrant IV True False
B The final image of ABC is a right triangle True False
C DeMarcus will get the same result if he True Falseperforms the reflection followed by the translation and rotation
Unit 9 1185
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3 cm
9 cmSALT
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hton Mifflin H
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pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
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Com
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Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
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oug
hton Mifflin H
arcourt Publishin
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pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
3 cm
9 cmSALT
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
5 A volleyball has a radius of about 8 inches A soccer ball has a radius of about 425 inches Determine whether each statement regarding the volume of each ball is true Select True or False for each statement
A The volume of the volleyball is about 6827π in3 True False
B The volume of the soccer ball is about 768π in3 True False
C The volume of the volleyball is about 375π times the volume of the soccer ball True False
6 A cone and a cylinder have the same height and base diameter Is each statement regarding the volume of each shape true Select True or False for each statement
A If the height is 8 cm and the base diameter is 6 cm the volume of the cone is 72π cm3 True False
B If the height is 6 cm and the base diameter is 4 cm the volume of the cylinder is 24π cm3 True False
C The volume of the cylinder is always 3 times the volume of the cone True False
7 A vase is in the shape of a cylinder with a height of 15 inches The vase holds 375π in3 of water What is the diameter of the base of the vase Show your work
8 A salt shaker is a cylinder with half a sphere on top The radius of the base of the salt shaker is 3 cm and the height of the cylindrical bottom is 9 cm as shown in the diagram
What is the volume of the salt shaker Explain how you got your answer
9 A cube is dilated by a factor of 4 By what factor does its volume increase Explain your reasoning
Unit 9 1186
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton
Mif
flin
Har
cour
t Pub
lishi
ng
Com
pan
y
Performance Tasks10 A scientist wants to compare the volumes of two cylinders One is twice as
high and has a diameter two times as long as the other If the volume of the smaller cylinder is 30 c m 3 what is the volume of the larger cylinder
11 You are trying to pack in preparation for a trip and need to fit a collection of childrenrsquos toys in a box Each individual toy is a composite figure of four cubes and all of the toys are shown in the figure Arrange the toys in an orderly fashion so that they will fit in the smallest box possible Draw the arrangement What is the volume of the box if each of the cubes have side lengths of 10 cm
12 A carpenter has a wooden cone with a slant height of 16 inches and a diameter of 12 inches The vertex of the cone is directly above the center of its base He measures halfway down the slant height and makes a cut parallel to the base He now has a truncated cone and a cone half the height of the original
A He expected the two parts to weigh about the same but they donrsquot Which is heavier Why
B Find the ratio of the weight of the small cone to that of the
truncated cone Show your work
Unit 9 1187
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D
copy H
oug
hton Mifflin H
arcourt Publishin
g Com
pany
mAth in CAreers
model maker A jewelry maker creates a pendant out of glass by attaching two square-based pyramids at their bases to form an octahedron Each triangular face of the octahedron is an equilateral triangle
a Derive a formula for the volume of the pendant if the side length is a Show your work
b The jewelry maker wants to package the pendant in a cylindrical box What should be the smallest dimensions of the box if the pendant just fits inside in terms of a Explain how you determined your answer
c What is the volume of empty space inside the box Your answer should be in terms of a and rounded to two decimal places Show your work
Unit 9 1188
DO NOT EDIT--Changes must be made through File info CorrectionKey=NL-DCA-D