Surface Area and Volume of Spheres

Transcription

1 12.6 Suface Aea and Volume of Sphees Befoe You found suface aeas and volumes of polyheda. Now You will find suface aeas and volumes of sphees. Why? So you can find the volume of a tennis ball, as in Ex. 33. Key Vocabulay sphee cente, adius, chod, diamete geat cicle hemisphees A sphee is the set of all points in space equidistant fom a given point. This point is called the cente of the sphee. A adius of a sphee is a segment fom the cente to a point on the sphee. A chod of a sphee is a segment whose endpoints ae on the sphee. A diamete of a sphee is a chod that contains the cente. cente C adius chod C diamete As with cicles, the tems adius and diamete also epesent distances, and the diamete is twice the adius. THEOREM THEOREM Suface Aea of a Sphee The suface aea S of a sphee is S 5 4π 2, whee is the adius of the sphee. Fo You Notebook S 5 4p 2 USE FORMULAS If you undestand how a fomula is deived, then it will be easie fo you to emembe the fomula. SURFACE AREA FORMULA To undestand how the fomula fo the suface aea of a sphee is deived, think of a baseball. The suface aea of a baseball is sewn fom two conguent shapes, each of which esembles two joined cicles, as shown. So, the entie coveing of the baseball consists of fou cicles, each with adius. The aea A of a cicle with adius is A 5π 2. So, the aea of the coveing can be appoximated by 4π 2. This is the fomula fo the suface aea of a sphee. leathe coveing 838 Chapte 12 Suface Aea and Volume of Solids

2 E X A M P L E 1 Find the suface aea of a sphee Find the suface aea of the sphee. 8 in. S 5 4π 2 Fomula fo suface aea of a sphee 5 4π(8 2 ) Substitute 8 fo π Simplify. ø Use a calculato. c The suface aea of the sphee is about squae inches. E X A M P L E 2 Standadized Test Pactice The suface aea of the sphee is 20.25p squae centimetes. What is the diamete of the sphee? A 2.25 cm C 5.5 cm B 4.5 cm D cm S p cm 2 S 5 4π 2 Fomula fo suface aea of a sphee 20.25π 5 4π 2 Substitute 20.25p fo S Divide each side by 4p. AVOID ERRORS Be sue to multiply the value of by 2 to find the diamete Find the positive squae oot. The diamete of the sphee is 2 5 2p centimetes. c The coect answe is B.ABCD GUIDED PRACTICE fo Examples 1 and 2 1. The diamete of a sphee is 40 feet. Find the suface aea of the sphee. 2. The suface aea of a sphee is 30π squae metes. Find the adius of the sphee. GREAT CIRCLES If a plane intesects a sphee, the intesection is eithe a single point o a cicle. If the plane contains the cente of the sphee, then the intesection is a geat cicle of the sphee. The cicumfeence of a geat cicle is the cicumfeence of the sphee. Evey geat cicle of a sphee sepaates the sphee into two conguent halves called hemisphees. geat cicle hemisphees 12.6 Suface Aea and Volume of Sphees 839

3 E X A M P L E 3 Use the cicumfeence of a sphee EXTREME SPORTS In a spot called spheeing, a peson olls down a hill inside an inflatable ball suounded by anothe ball. The diamete of the oute ball is 12 feet. Find the suface aea of the oute ball. The diamete of the oute sphee is 12 feet, so the adius is 12 } feet. Use the fomula fo the suface aea of a sphee. S 5 4π 2 5 4π(6 2 ) 5 144π c The suface aea of the oute ball is 144π, o about squae feet. GUIDED PRACTICE fo Example 3 3. In Example 3, the cicumfeence of the inne ball is 6π feet. Find the suface aea of the inne ball. Round you answe to two decimal places. VOLUME FORMULA Imagine that the inteio of a sphee with adius is appoximated by n pyamids, each with a base aea of B and a height of. The volume of each pyamid is 1 } 3 B and the sum of the base aeas is nb. The suface aea of the sphee is appoximately equal to nb, o 4π 2. So, you can appoximate the volume V of the sphee as follows. V ø n1 1 } 3 B2 Each pyamid has a volume of 1 } 3 B. ø 1 } 3 (nb) Regoup factos. 5 1 } 3 (4π 2 ) Substitute 4p 2 fo nb. Aea 5 B 5 4 } 3 π 3 Simplify. THEOREM THEOREM Volume of a Sphee The volume V of a sphee is V 5 } 4 π 3, 3 whee is the adius of the sphee. Fo You Notebook V 5 4 } 3 p Chapte 12 Suface Aea and Volume of Solids

4 E X A M P L E 4 Find the volume of a sphee The socce ball has a diamete of 9 inches. Find its volume. The diamete of the ball is 9 inches, so the adius is 9 } inches. V 5 4 } 3 π 3 Fomula fo volume of a sphee 5 } 4 π(4.5) 3 Substitute π Simplify. ø Use a calculato. c The volume of the socce ball is 121.5π, o about cubic inches. E X A M P L E 5 Find the volume of a composite solid Find the volume of the composite solid. Volume of solid 5 Volume of cylinde 2 Volume of hemisphee 2 in. 2 in. 5π 2 h 2 1 } } 3 π 3 2 5π(2) 2 (2) 2 2 } 3 π(2) 3 Fomulas fo volume Substitute. 5 8π 2 2 } 3 (8π) Multiply } 3 π 2 16 } 3 π Rewite factions using least common denominato. 5 8 } 3 π Simplify. c The volume of the solid is 8 } 3 π, o about 8.38 cubic inches. at classzone.com GUIDED PRACTICE fo Examples 4 and 5 4. The adius of a sphee is 5 yads. Find the volume of the sphee. Round you answe to two decimal places. 5. A solid consists of a hemisphee of adius 1 mete on top of a cone with the same adius and height 5 metes. Find the volume of the solid. Round you answe to two decimal places Suface Aea and Volume of Sphees 841

5 12.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 fo Exs. 3, 13, and 31 5 STANDARDIZED TEST PRACTICE Exs. 2, 6, 20, 28, 33, and VOCABULARY What ae the fomulas fo finding the suface aea of a sphee and the volume of a sphee? 2. WRITING When a plane intesects a sphee, what point in the sphee must the plane contain fo the intesection to be a geat cicle? Explain. EXAMPLE 1 on p. 839 fo Exs. 3 5 FINDING SURFACE AREA Find the suface aea of the sphee. Round you answe to two decimal places ft cm m EXAMPLE 2 on p. 839 fo Ex. 6 EXAMPLE 3 on p. 840 fo Exs MULTIPLE CHOICE What is the appoximate adius of a sphee with suface aea 32π squae metes? A 2 metes B 2.83 metes C 4.90 metes D 8 metes USING A GREAT CIRCLE In Execises 7 9, use the sphee below. The cente of the sphee is C and its cicumfeence is 9.6p inches. 7. Find the adius of the sphee. 8. Find the diamete of the sphee. 9. Find the suface aea of one hemisphee. C 10. ERROR ANALYSIS Descibe and coect the eo in finding the suface aea of a hemisphee with adius 5 feet. S 5 4π 2 5 4π(5) π ø ft 2 5 ft 11. GREAT CIRCLE The cicumfeence of a geat cicle of a sphee is 48.4π centimetes. What is the suface aea of the sphee? EXAMPLE 4 on p. 841 fo Exs FINDING VOLUME Find the volume of the sphee using the given adius o diamete d. Round you answe to two decimal places in mm 14. d 5 5 cm 842 Chapte 12 Suface Aea and Volume of Solids

6 15. ERROR ANALYSIS Descibe and coect the eo in finding the volume of a sphee with diamete 16 feet. V 5 4 } 3 π } 3 π(8) π ø ft 2 USING VOLUME In Execises 16 18, find the adius of a sphee with the given volume V. Round you answes to two decimal places. 16. V m V cm V 5 20, in FINDING A DIAMETER The volume of a sphee is 36π cubic feet. What is the diamete of the sphee? 20. MULTIPLE CHOICE Let V be the volume of a sphee, S be the suface aea of the sphee, and be the adius of the sphee. Which equation epesents the elationship between these thee measues? A V 5 S } 3 B V 5 2 S } 3 C V 5 3 } 2 S D V 5 3 } 2 2 S EXAMPLE 5 on p. 841 fo Exs COMPOSITE SOLIDS Find the suface aea and the volume of the solid. The cylindes and cones ae ight. Round you answe to two decimal places cm 7 in. 14 ft 5.8 ft 12.6 cm 3.3 in. USING A TABLE Copy and complete the table below. Leave you answes in tems of p Radius of sphee Cicumfeence of geat cicle Suface aea of sphee Volume of sphee 10 ft???? 26π in.???? 2500π cm 2???? 12,348π m MULTIPLE CHOICE A sphee is inscibed in a cube of volume 64 cubic centimetes. What is the suface aea of the sphee? A 4π cm 2 B 32 } 3 π cm 2 C 16π cm 2 D 64π cm CHALLENGE The volume of a ight cylinde is the same as the volume of a sphee. The adius of the sphee is 1 inch. a. Give thee possibilities fo the dimensions of the cylinde. b. Show that the suface aea of the cylinde is sometimes geate than the suface aea of the sphee Suface Aea and Volume of Sphees 843

7 PROBLEM SOLVING EXAMPLE 5 on p. 841 fo Ex GRAIN SILO A gain silo has the dimensions shown. The top of the silo is a hemispheical shape. Find the volume of the gain silo. 31. GEOGRAPHY The cicumfeence of Eath is about 24,855 miles. Find the suface aea of the Westen Hemisphee of Eath. 60 ft 20 ft 32. MULTI-STEP PROBLEM A ball has volume cubic centimetes. a. Find the adius of the ball. Round you answe to two decimal places. b. Find the suface aea of the ball. Round you answe to two decimal places. 33. SHORT RESPONSE Tennis balls ae stoed in a cylindical containe with height inches and adius 1.43 inches. a. The cicumfeence of a tennis ball is 8 inches. Find the volume of a tennis ball. b. Thee ae 3 tennis balls in the containe. Find the amount of space within the cylinde not taken up by the tennis balls. 34. EXTENDED RESPONSE A patially filled balloon has cicumfeence 27π centimetes. Assume the balloon is a sphee. a. Calculate Find the volume of the balloon. b. Pedict Suppose you double the adius by inceasing the ai in the balloon. Explain what you expect to happen to the volume. c. Justify Find the volume of the balloon with the adius doubled. Was you pediction fom pat (b) coect? What is the atio of this volume to the oiginal volume? 35. GEOGRAPHY The Toid Zone on Eath is the aea between the Topic of Cance and the Topic of Capicon, as shown. The distance between these two topics is about 3250 miles. You can think of this distance as the height of a cylindical belt aound Eath at the equato, as shown. a. Estimate the suface aea of the Toid Zone and the suface aea of Eath. (Eath s adius is about 3963 miles at the equato.) b. A meteoite is equally likely to hit anywhee on Eath. Estimate the pobability that a meteoite will land in the Toid Zone. equato Topic of Cance Toid Zone Topic of Capicon 3250 mi WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST PRACTICE

8 36. REASONING List the following thee solids in ode of (a) suface aea, and (b) volume, fom least to geatest. Solid I Solid II Solid III ROTATION A cicle with diamete 18 inches is otated about its diamete. Find the suface aea and the volume of the solid fomed. 38. TECHNOLOGY A cylinde with height 2x is inscibed in a sphee with adius 8 metes. The cente of the sphee is the midpoint of the altitude that joins the centes of the bases of the cylinde. a. Show that the volume V of the cylinde is 2πx(64 2 x 2 ). b. Use a gaphing calculato to gaph V 5 2πx(64 2 x 2 ) fo values of x between 0 and 8. Find the value of x that gives the maximum value of V. c. Use the value fo x fom pat (b) to find the maximum volume of the cylinde. 8 m 39. CHALLENGE A sphee with adius 2 centimetes is inscibed in a ight cone with height 6 centimetes. Find the suface aea and the volume of the cone. MIXED REVIEW PREVIEW Pepae fo Lesson 12.7 in Exs In Execises 40 and 41, the polygons ae simila. Find the atio (ed to blue) of thei aeas. Find the unknown aea. Round you answe to two decimal places. (p. 737) 40. Aea of n ABC 5 42 ft Aea of PQRS cm 2 Aea of n DEF 5? Aea of JKLM 5? A 8 ft C B D 5 ft F E J M 15 cm K P P L S 21 cm R Find the pobability that a andomly chosen point in the figue lies in the shaded egion. (p. 771) A cone is inscibed in a ight cylinde with volume 330 cubic units. Find the volume of the cone. (pp. 819, 829) 9.7 EXTRA PRACTICE fo Lesson 12.6, p. 919 ONLINE QUIZ at classzone.com 845