ExerciseExerciseIf a pyramid and a cone have bases with the same area and altitudes that are equal, are their surface areas equal?

If a pyramid and a cone have bases with the same area and altitudes that are equal, are their surface areas equal?

nono

If a pyramid and a cone have bases with the same area and altitudes that are equal, are their volumes equal?

If a pyramid and a cone have bases with the same area and altitudes that are equal, are their volumes equal?

yesyes

ExerciseExercise

In this text, what is the difference between h and H?In this text, what is the difference between h and H?

h = length of the altitude of a plane figure and H = length of the altitude of a solid figure.

h = length of the altitude of a plane figure and H = length of the altitude of a solid figure.

ExerciseExercise

What would the calculation of bhH give?

What would the calculation of bhH give?

the volume of a triangular prismthe volume of a triangular prism

1212

ExerciseExercise

hh

wwll

hh

wwll

Formula: Volume of a Pyramid or a ConeFormula: Volume of a Pyramid or a ConeV = BH The volume of a pyramid or cone (V) is equal to one- third the area of the base (B) times

the height (H).

V = BH The volume of a pyramid or cone (V) is equal to one- third the area of the base (B) times

the height (H).

1313

Find the volume of the square pyramid.Find the volume of the square pyramid.

= 256 cm3= 256 cm3

V = BHV = BH1313

= (82)(12)= (82)(12)1313

8 cm8 cm 8 cm8 cm

12 cm12 cm

Example 1Example 1

Find the volume of the cone.Find the volume of the cone.

≈ 100.5 cm3≈ 100.5 cm3

V = BHV = BH1313

= p(42)(6)= p(42)(6)1313

= 32p= 32p

= 32(3.14)= 32(3.14)

6 cm6 cm

4 cm4 cm

Example 2Example 2

What is the volume of a pyramid if its height is 10 units and its base is 8 units by 12 units?

What is the volume of a pyramid if its height is 10 units and its base is 8 units by 12 units?

320 units3320 units3

ExampleExample

What would happen to the volume of the pyramid in the previous question if its length were doubled?

What would happen to the volume of the pyramid in the previous question if its length were doubled?

The volume would be doubled.

The volume would be doubled.

ExampleExample

What would happen to the volume if any single dimension were doubled?

What would happen to the volume if any single dimension were doubled?

ExampleExample

The volume would be doubled.

The volume would be doubled.

What would happen if all the dimensions were doubled?What would happen if all the dimensions were doubled?

The volume would be multiplied by a factor of 23 = 8.

The volume would be multiplied by a factor of 23 = 8.

ExampleExample

What is the volume of a square pyramid if each side of its base is 6 units and its height is 5 units?

What is the volume of a square pyramid if each side of its base is 6 units and its height is 5 units?

60 units360 units3

ExampleExample

What would happen to the volume of the pyramid in the previous question if the sides of the square base were doubled?

What would happen to the volume of the pyramid in the previous question if the sides of the square base were doubled?

The volume would be multiplied by a factor of 22 = 4.

The volume would be multiplied by a factor of 22 = 4.

ExampleExample

Formula: Volume of a SphereFormula: Volume of a SphereV = pr3 The volume of a

sphere (V) is equal to the product of , p, and the radius cubed (r).

V = pr3 The volume of a sphere (V) is equal to the product of , p, and the radius cubed (r).

4343

4343

Find the volume of a sphere with a diameter of 15 ft. to the nearest hundredth. Find the number of gallons it will hold. (1 ft.3 = 7.48 gal.)

Find the volume of a sphere with a diameter of 15 ft. to the nearest hundredth. Find the number of gallons it will hold. (1 ft.3 = 7.48 gal.)

r = = 7.5 ft.r = = 7.5 ft.152

152

Example 3Example 3

V = pr3V = pr34343

= p(7.53)= p(7.53)4343

= p(421.875)= p(421.875)4343

= p= p1,687.53

1,687.53

≈ 1,766.25 ft.3≈ 1,766.25 ft.3

Example 3Example 3

≈ 13,212 gal.≈ 13,212 gal.7.48(1,766.25)7.48(1,766.25)

Example 3Example 3

Find the radius of a sphere with a volume of 288p m3.Find the radius of a sphere with a volume of 288p m3.

V = pr3V = pr34343

pr3 = 288ppr3 = 288p4343

pr3 = (288p)pr3 = (288p)4343

3434 ( )( ) 3

434

Example 4Example 4

pr3 = 216ppr3 = 216p

r3 = 216r3 = 216

r = 6 mr = 6 m

Example 4Example 4

What is the volume of a sphere with a radius of 6 units?

What is the volume of a sphere with a radius of 6 units?

ExampleExample

904.32 units3904.32 units3

A city needs a 10,000 m3 water tower for its increasing population. What should the radius be if the water tower is in the form of a sphere?

A city needs a 10,000 m3 water tower for its increasing population. What should the radius be if the water tower is in the form of a sphere?

ExampleExample

13.37 m13.37 m

A grain storage bin is a steel cylinder with a conical top. One company markets a bin that is 18’ in diameter, 16’ high at the eaves, and 21’ high at the peak.

A grain storage bin is a steel cylinder with a conical top. One company markets a bin that is 18’ in diameter, 16’ high at the eaves, and 21’ high at the peak.

ExerciseExercise

What is the maximum number of bushels of wheat (rounded to the nearest bushel) that can be stored in the bin? There are 0.8 bushels in one cubic foot.

What is the maximum number of bushels of wheat (rounded to the nearest bushel) that can be stored in the bin? There are 0.8 bushels in one cubic foot.

ExerciseExercise

V = pr2H + pr2HV = pr2H + pr2H1313

= 1,296p + 135p = 1,431p ft.3= 1,296p + 135p = 1,431p ft.3

= p(92)(16) + p(92)(5) = p(92)(16) + p(92)(5) 1313

= 1,431p ft.3= 1,431p ft.3 0.8 bu.1 ft.3

0.8 bu.1 ft.3( )( )

≈ 3,595 bu.≈ 3,595 bu.

ExerciseExercise