16.4 Volume of Spheres

Transcription

1 Name Class Date 16.4 Volume of Sphees Essential Question: How can you use the fomula fo the volume of a sphee to calculate the volumes of composite figues? Exploe G.11.D Apply the fomulas fo the volume of thee-dimensional figues, including... sphees,... to solve poblems using appopiate units of measue. Developing a Volume Fomula Resouce Locke To find the volume of a sphee, compae one of its hemisphees to a cylinde of the same height and adius fom which a cone has been emoved. The egion of a plane that intesects a solid figue is called a coss section. To show that coss sections have the same aea at evey level, use the Pythagoean Theoem to find a elationship between, x, and R. R x A coss section of the cylinde with the cone emoved is a ing. To find the aea of the ing, find the aea of the oute cicle and of the inne cicle. Then subtact the aea of the inne cicle fom the oute cicle. x x Module Lesson 4

2 C Find an expession fo the volume of the cylinde with the cone emoved. D Use Cavaliei s pinciple to deduce the volume of a sphee with adius. Reflect 1. How do you know that the height h of the cylinde with the cone emoved is equal to the adius? 2. What happens to the coss-sectional aeas when x = 0? when x =? Module Lesson 4

3 Explain 1 Finding the Volume of a Sphee The elationship you discoveed in the Exploe can be stated as a volume fomula. Volume of a Sphee The volume of a sphee with adius is given by V = 4_ 3 π 3. You can use a fomula fo the volume of a sphee to solve poblems involving volume and capacity. Example 1 The figue epesents a spheical helium-filled balloon. This touist attaction allows up to 28 passenges at a time to ide in a gondola suspended undeneath the balloon, as it cuises at an altitude of 500 ft. How much helium, to the neaest hunded gallons, does the balloon hold? Round to the neaest tenth. (Hint: 1 gal ft 3 ) Step 1 Find the adius of the balloon. The adius is half of the diamete, so = 1_ (72 ft) = 36 ft. 2 Step 2 Find the volume of the balloon in cubic feet. 72 ft You Tun V = = 4_ 3 π 3 3 4_ ( ) 3 π ft 3 Step 3 Find the capacity of the balloon to the neaest gallon. ft 3 ft 3 1 gal _ ft 3 gal A spheical wate tank has a diamete of 27 m. How much wate can the tank hold, to the neaest lite? (Hint: 1,000 L = 1 m 3 ) 3. Find the volume of the tank in cubic metes. 4. Find the capacity of the tank to the neaest lite. Module Lesson 4

4 Explain 2 Finding the Volume of a Composite Figue You can find the volume of a composite figue using appopiate volume fomulas fo the diffeent pats of the figue. Example 2 Find the volume of the composite figue. Round to the neaest cubic centimete. Step 1 Find the volume of the hemisphee. 5 cm 13 cm Step 2 Find the height of the cone. h 2 + ( ) h 2 + = h 2 = h = 2 = ( ) 2 Step 3 Find the volume of the cone. The cone has the same adius as the hemisphee, = V = = 1_ 3 π 2 h 2 1_ 3 ( ) π cm. ( ) = cm 3 Step 4 Find the total volume. Total volume = volume of hemisphee + volume of cone 3 3 = cm + cm 3 cm Reflect 5. Is it possible to ceate a figue by taking a cone and emoving fom it a hemisphee with the same adius? You Tun 6. A composite figue is a cylinde with a hemispheical hole in the top. The bottom of the hemisphee is tangent to the base of the cylinde. Find the volume of the figue, to the neaest tenth. 7 in. Module Lesson 4

5 Elaboate 7. Discussion Could you use an inscibed pism to deive the volume of a hemisphee? Why o why not? Ae thee any othe ways you could appoximate a hemisphee, and what poblems would you encounte in finding its volume? 8. Essential Question Check-In A gumball is in the shape of a sphee, with a spheical hole in the cente. How might you calculate the volume of the gumball? What measuements ae needed? Evaluate: Homewok and Pactice 1. Analyze Relationships Use the diagam of a sphee inscibed in a cylinde to descibe the elationship between the volume of a sphee and the volume of a cylinde. Online Homewok Hints and Help Exta Pactice Module Lesson 4

6 Find the volume of the sphee. Round the answe to the neaest tenth in. 11 ft Cicumfeence of geat cicle is 14π cm Find the volume of the sphee. Leave the answe in tems of π cm 1 m A = 81π in 2 Module Lesson 4

7 Find the volume of the composite figue. Leave the answe in tems of π ft 2 ft 2 in. 8 in. 3 in. Find the volume of the composite figue. Round the answe to the neaest tenth cm 10 mm 24 mm 4 cm 10 cm 8 cm 8 mm Module Lesson 4

8 12. Analyze Relationships Appoximately how many times as geat is the volume of a gapefuit with diamete 10 cm as the volume of a lime with diamete 5 cm? 13. A bead is fomed by dilling a cylindical hole with a 2 mm diamete though a sphee with an 8 mm diamete. Estimate the volume of the bead to the neaest whole. 14. Algeba Wite an expession epesenting the volume of the composite figue fomed by a hemisphee with adius and a cube with side length 2. Image Cedits: Jasmina81/iStockPhoto.com Module Lesson 4

9 15. One gallon of popane yields appoximately 91,500 BTU. About how many BTUs does the spheical stoage tank shown povide? Round to the neaest billion BTUs. (Hint: 1 f t gal) 5 ft 16. The aquaium shown is a ectangula pism that is filled with wate. You dop a spheical ball with a diamete of 6 inches into the aquaium. The ball sinks, causing the wate to spill fom the tank. How much wate is left in the tank? Expess you answe to the neaest tenth. (Hint: 1 in gal) 20 in. 12 in. 12 in. 17. A sphee with diamete 8 cm is inscibed in a cube. Find the atio of the volume of the cube to the volume of the sphee. A. 6_ π B. 2_ 3π C. 3π_ 4 D. 3π_ 2 Module Lesson 4

10 Fo Execises 18 20, use the table. Round each volume to the neaest billion π. Planet Diamete (mi) Mecuy 3,032 Venus 7,521 Eath 7,926 Mas 4,222 Jupite 88,846 Satun 74,898 Uanus 31,763 Neptune 30, Explain the Eo Magaet used the mathematics shown to find the volume of Satun. V = 4_ 3 π 2 = 4_ 4_ π (6,000,000,000) 8,000,000,000π 3 3 Explain the two eos Magaet made, then give the coect answe. 19. The sum of the volumes of Venus and Mas is about equal to the volume of which planet? 20. How many times as geat as the volume of the smallest planet is the volume of the lagest planet? Round to the neaest thousand. Module Lesson 4

11 H.O.T. Focus on Highe Ode Thinking 21. Make a Conjectue The bathysphee was an ealy vesion of a submaine, invented in the 1930s. The inside diamete of the bathysphee was 54 inches, and the steel used to make the sphee was 1.5 inches thick. It had thee 8-inch diamete windows. Estimate the volume of steel used to make the bathysphee. 22. Explain the Eo A student solved the poblem shown. Explain the student s eo and give the coect answe to the poblem. A spheical gasoline tank has a adius of 0.5 ft. When filled, the tank povides 446,483 BTU. How many BTUs does one gallon of gasoline yield? Round to the neaest thousand BTUs and use the fact that 1 ft gal. The volume of the tank is 4 3 π 3 = 4 3 π (0.5) 3 ft 3. Multiplying by 7.48 shows that this is appoximately 3.92 gal. So the numbe of BTUs in one gallon of gasoline is appoximately 446, ,750,000 BTU. Image Cedits: Regien Paassen/Shuttestock 23. Pesevee in Poblem Solving The top of a gumball machine is an 18 in. sphee. The machine holds a maximum of 3300 gumballs, which leaves about 43% of the space in the machine empty. Estimate the diamete of each gumball. Module Lesson 4

12 Lesson Pefomance Task Fo his science poject, Bizbo has decided to build a scale model of the sola system. He stats with a gapefuit with a adius of 2 inches to epesent Eath. His Eath weighs 0.5 pounds. Find each of the following fo Bizbo s model. Use the ounded figues in the table. Round you answes to two significant figues. Use 3.14 fo π. 1. the scale of Bizbo s model: 1 inch = miles 2. Eath s distance fom the Sun, in inches and in miles 3. Neptune s distance fom the Sun, in inches and in miles 4. the Sun s volume, in cubic inches and cubic feet 5. the Sun s weight, in pounds and in tons (Note: the Sun s density is 0.26 times the Eath s density.) Module Lesson 4