Holt Geometry 10-8 Spheres Learn and apply the formula for the volume of a sphere. Learn and apply...
-
Upload
marilynn-parsons -
Category
Documents
-
view
217 -
download
3
Transcript of Holt Geometry 10-8 Spheres Learn and apply the formula for the volume of a sphere. Learn and apply...
Holt Geometry
10-8 Spheres
Learn and apply the formula for the volume of a sphere.Learn and apply the formula for the surface area of a sphere.
Objectives
Holt Geometry
10-8 Spheres
spherecenter of a sphereradius of a spherehemispheregreat circle
Vocabulary
Holt Geometry
10-8 Spheres
A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres
Holt Geometry
10-8 Spheres
Holt Geometry
10-8 Spheres
Example 1A: Finding Volumes of Spheres
Find the volume of the sphere. Give your answer in terms of .
= 2304 in3 Simplify.
Volume of a sphere.
Holt Geometry
10-8 Spheres
Example 1B: Finding Volumes of Spheres
Find the diameter of a sphere with volume 36,000 cm3.
Substitute 36,000 for V.
27,000 = r3
r = 30
d = 60 cm d = 2r
Take the cube root of both sides.
Volume of a sphere.
Holt Geometry
10-8 Spheres
Example 1C: Finding Volumes of Spheres
Find the volume of the hemisphere.
Volume of a hemisphere
Substitute 15 for r.
= 2250 m3 Simplify.
Holt Geometry
10-8 Spheres
Check It Out! Example 1
Find the radius of a sphere with volume 2304 ft3.
Volume of a sphere
Substitute for V.
r = 12 ft Simplify.
Holt Geometry
10-8 Spheres
Example 2: Sports Application
A sporting goods store sells exercise balls in two sizes, standard (22-in. diameter) and jumbo (34-in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball?
standard ball: jumbo ball:
A jumbo ball is about 3.7 times as great in volume as a standard ball.
Holt Geometry
10-8 Spheres
Holt Geometry
10-8 Spheres
Example 3A: Finding Surface Area of Spheres
Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .
S = 4r2
S = 4(38)2 = 5776 cm2
Surface area of a sphere
Holt Geometry
10-8 Spheres
Example 3B: Finding Surface Area of Spheres
Find the volume of a sphere with surface area 324 in2. Give your answers in terms of .
Substitute 324 for S.324 = 4r2
r = 9 Solve for r.
Substitute 9 for r.
The volume of the sphere is 972 in2.
S = 4r2 Surface area of a sphere
Holt Geometry
10-8 Spheres
Example 3C: Finding Surface Area of Spheres
Find the surface area of a sphere with a great circle that has an area of 49 mi2.
Substitute 49 for A.49 = r2
r = 7 Solve for r.
S = 4r2
= 4(7)2 = 196 mi2 Substitute 7 for r.
A = r2 Area of a circle
Holt Geometry
10-8 Spheres
Check It Out! Example 3
Find the surface area of the sphere.
Substitute 25 for r.
S = 2500 cm2
S = 4r2
S = 4(25)2
Surface area of a sphere
Holt Geometry
10-8 Spheres
Example 4: Exploring Effects of Changing Dimensions
The radius of the sphere is
multiplied by . Describe the
effect on the volume.
original dimensions: radius multiplied by :
Notice that . If the radius is multiplied by
, the volume is multiplied by , or .
Holt Geometry
10-8 Spheres
Check It Out! Example 4
The radius of the sphere is divided by 3. Describe the effect on the surface area.
original dimensions: dimensions divided by 3:
The surface area is divided by 9.
S = 4r2
= 4(3)2 = 36 m3
S = 4r2
= 4(1)2 = 4 m3
Holt Geometry
10-8 Spheres
Example 5: Finding Surface Areas and Volumes of Composite Figures
Find the surface area and volume of the composite figure. Give your answer in terms of .
Step 1 Find the surface area of the composite figure.
The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.
Holt Geometry
10-8 Spheres
Example 5 Continued
The surface area of the composite figure is
L(cylinder) = 2rh = 2(6)(9) = 108 in2
B(cylinder) = r2 = (6)2 = 36 in2
72 + 108 + 36 = 216 in2.
Find the surface area and volume of the composite figure. Give your answer in terms of .
Holt Geometry
10-8 Spheres
Step 2 Find the volume of the composite figure.
Example 5 Continued
Find the surface area and volume of the composite figure. Give your answer in terms of .
The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cylinder.
The volume of the composite figure is 144 + 324 = 468 in3.
Holt Geometry
10-8 Spheres
Check It Out! Example 5
Find the surface area and volume of the composite figure.
Step 1 Find the surface area of the composite figure.
The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.
Holt Geometry
10-8 Spheres
Check It Out! Example 5 Continued
The surface area of the composite figure is
Find the surface area and volume of the composite figure.
L(cylinder) = 2rh = 2(3)(5) = 30 ft2
B(cylinder) = r2 = (3)2 = 9 ft2
18 + 30 + 9 = 57 ft2.
Holt Geometry
10-8 Spheres
Step 2 Find the volume of the composite figure.
Find the surface area and volume of the composite figure.
Check It Out! Example 5 Continued
The volume of the composite figure is the volume of the cylinder minus the volume of the hemisphere.
V = 45 – 18 = 27 ft3