9.3 Volume of Spheres
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1 ? LESSON 9. Volume of Sphees ESSENTIAL QUESTION How do you find the volume of a sphee? Expessions, equations, and elationships Solve poblems involving the volume of sphees. EXPLORE ACTIVITY Modeling the Volume of a Sphee A sphee is a thee-dimensional figue with all points the same distance fom the cente. The adius of a sphee is the distance fom the cente to any point on the sphee. You have seen that a cone fills 1_ of a cylinde of the same adius and height h. If you wee to do a simila expeiment with a sphee of the same adius, you would find that a sphee fills 2_ of the cylinde. The cylinde s height is equal to twice the adius of the sphee. h = 2 h = 2 h = 2 STEP 1 Wite the fomula V = Bh fo each shape. Use B = π 2 and substitute the factions you know fo the cone and sphee. Houghton Mifflin Hacout Publishing Company STEP 2 STEP Cylinde Cone Sphee V = π 2 h V = 1 π 2 h V = 2 π 2 h Notice that a sphee always has a height equal to twice the adius. Substitute 2 fo h. V = 2 π 2 (2) Simplify this fomula fo the volume of a sphee. V = π Reflect 1. Analyze Relationships A cone has a adius of and a height of 2. A sphee has a adius of. Compae the volume of the sphee and cone. Lesson
2 Math On the Spot my.hw.com Finding the Volume of a Sphee Using a Fomula The Exploe Activity illustates a fomula fo the volume of a sphee with adius. Volume of a Sphee The volume V of a sphee is 4_ π times the cube of the adius. V = 4_ π EXAMPLE 1 Find the volume of each sphee. Round you answes to the neaest tenth if necessay. Use.14 fo π. Math Talk Mathematical Pocesses If you know the diamete of a sphee, how would the fomula fo the volume of a sphee be witten in tems of d? A B 2.1 cm 7 cm V = 4_ π 4_ _ The volume is about 8.8 c m. Since the diamete is 7 cm, the adius is.5 cm. V = 4_ π Substitute. Simplify. Multiply. Pesonal Math Taine Online Assessment and Intevention my.hw.com YOUR TURN The volume is about c m. 4_ _ Find the volume of each sphee. Round you answes to the neaest tenth. Use.14 fo π. 2. A sphee has a adius of 10 centimetes.. A sphee has a diamete of.4 metes. Substitute. Simplify. Multiply. Houghton Mifflin Hacout Publishing Company 258 Unit
3 Finding the Volume of a Sphee in a Real-Wold Context Many spots, including golf and tennis, use a ball that is spheical in shape. EXAMPLE 2 Math On the Spot my.hw.com Socce balls come in seveal diffeent sizes. One socce ball has a diamete of 22 centimetes. What is the volume of this socce ball? Round you answe to the neaest tenth. Use.14 fo π. STEP 1 Find the adius. = d 2 = 11 cm STEP 2 Find the volume of the socce ball. V = 4_ π 4_ _ Substitute. Simplify. Multiply. The volume of the socce ball is about c m. Houghton Mifflin Hacout Publishing Company Image Cedits: PhotoDisc/ Getty Images Reflect 4. What is the volume of the socce ball in tems of π, to the neaest whole numbe multiple? Explain you answe. 5. Analyze Relationships The diamete of a basketball is about 1.1 times that of a socce ball. The diamete of a tennis ball is about 0. times that of a socce ball. How do the volumes of these balls compae to that of a socce ball? Explain. YOUR TURN 6. Val measues the diamete of a ball as 12 inches. How many cubic inches of ai does this ball hold, to the neaest tenth? Use.14 fo π. Pesonal Math Taine Online Assessment and Intevention my.hw.com Lesson
4 Guided Pactice 1. Vocabulay A sphee is a thee-dimensional figue with all points fom the cente. (Exploe Activity) 2. Vocabulay The is the distance fom the cente of a sphee to a point on the sphee. (Exploe Activity) Find the volume of each sphee. Round you answes to the neaest tenth if necessay. Use.14 fo π. (Example 1) in. 20 cm 5. A sphee has a adius of 1.5 feet. 6. A sphee has a diamete of 2 yads. 7. A baseball has a diamete of 2.9 inches. Find the volume of the baseball. Round you answe to the neaest tenth if necessay. Use.14 fo π. (Example 2) 8. A basketball has a adius of 4.7 inches. What is its volume to the neaest cubic inch? Use.14 fo π. (Example 2) 9. A company is deciding whethe to package a ball in a cubic box o a cylindical box. In eithe case, the ball will touch the bottom, top, and sides. (Exploe Activity)? a. What potion of the space inside the cylindical box is empty? Explain. b. Find an expession fo the volume of the cubic box. c. About what potion of the space inside the cubic box is empty? Explain. ESSENTIAL QUESTION CHECK-IN 10. Explain the steps you use to find the volume of a sphee. Houghton Mifflin Hacout Publishing Company 260 Unit
5 Name Class Date 9. Independent Pactice Find the volume of each sphee. Round you answes to the neaest tenth if necessay. Use.14 fo π. 11. adius of.1 metes 12. diamete of 18 inches my.hw.com Pesonal Math Taine Online Assessment and Intevention 19. Fossilized spheical eggs of dinosaus called titanosauid sauopods wee found in Patagonia. These eggs wee 15 centimetes in diamete. Find the volume of an egg. Round you answe to the neaest tenth. 1. = 6 in. 14. d = 6 m cm 20. Pesevee in Poblem Solving An ostich egg has about the same volume as a sphee with a diamete of 5 inches. If the eggshell is about 1 inch thick, find the 12 volume of just the shell, not including the inteio of the egg. Round you answe to the neaest tenth ft 21. Multistep Wite the steps you would use to find a fomula fo the volume of the figue at ight. Then wite the fomula. Houghton Mifflin Hacout Publishing Company The eggs of bids and othe animals come in many diffeent shapes and sizes. Eggs often have a shape that is nealy spheical. When this is tue, you can use the fomula fo a sphee to find thei volume. 17. The geen tutle lays eggs that ae appoximately spheical with an aveage diamete of 4.5 centimetes. Each tutle lays an aveage of 11 eggs at one time. Find the total volume of these eggs, to the neaest cubic centimete. 18. Hummingbids lay eggs that ae nealy spheical and about 1 centimete in diamete. Find the volume of an egg. Round you answe to the neaest tenth. Lesson
6 22. Citical Thinking Explain what happens to the volume of a sphee if you double the adius. 2. Multistep A cylindical can of tennis balls holds a stack of thee balls so that they touch the can at the top, bottom, and sides. The adius of each ball is 1.25 inches. Find the volume inside the can that is not taken up by the thee tennis balls. FOCUS ON HIGHER ORDER THINKING Wok Aea 24. Citique Reasoning A sphee has a adius of 4 inches, and a cube-shaped box has an edge length of 7.5 inches. J.D. says the box has a geate volume, so the sphee will fit in the box. Is he coect? Explain. 25. Citical Thinking Which would hold the most wate: a bowl in the shape of a hemisphee with adius, a cylindical glass with adius and height, o a cone-shaped dinking cup with adius and height? Explain. 26. Analyze Relationships Hai has models of a sphee, a cylinde, and a cone. The sphee s diamete and the cylinde s height ae the same, 2. The cylinde has adius. The cone has diamete 2 and height 2. Compae the volumes of the cone and the sphee to the volume of the cylinde. 27. A spheical helium balloon that is 8 feet in diamete can lift about 17 pounds. What does the diamete of a balloon need to be to lift a peson who weighs 16 pounds? Explain. Houghton Mifflin Hacout Publishing Company 262 Unit
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