Investigate Surface Area of Three-Dimensional Objects

Transcription

1 2.2 Suface Aea Blackfoot Cossing Exhibit Hall Focus on solving poblems involving the suface aea of theedimensional objects finding an unknown dimension of a theedimensional object given its suface aea Achitectual design ideas may evolve fom you cultue, icons, o eveyday life. The exhibit hall at the Blackfoot Cossing Histoical Pak in southen Albeta is a cultual, educational, and entetainment cente built at the site of the signing of Teaty 7. The knowledge of suface aea of integated stuctues is essential when constucting achitectual designs like the Blackfoot Exhibit Hall. Achitects use suface aea to calculate the amount of mateial needed. Investigate Suface Aea of Thee-Dimensional Objects Investigate Suface Aea of Thee-Dimensional Objects 1 Goup Definition of Suface Aea 2 Goup Stategy fo Detemining Suface Aea 1. As a goup, divide a sheet of pape into fou quadants. Label the quadants with the following: Quadant 1: Goup Definition of Suface Aea Quadant 2: Goup Stategy fo Detemining Suface Aea Quadant 3: Detemine Suface Aea Quadant 4: Altenative Method fo Detemining Suface Aea 3 Detemine Suface Aea 4 Altenative Method fo Detemining Suface Aea 66 MHR Chapte 2

2 2. As a goup, choose one of the thee-dimensional objects shown below. 4.2 cm 6 in. 10 cm 9 in. Complete quadant 1 and one of the thee emaining quadants. Then, pass the sheet on to anothe goup. 8 in. 3. a) Review and discuss the wok of the othe goups. b) Choose and complete one of the emaining quadants. Then, pass the sheet of pape on to anothe goup. 4. Repeat the pocess in step 3 fo the emaining quadant. 5. Reflect and Respond a) Did anothe goup s stategy wok bette than yous? Why o why not? b) Did you see any altenative stategies that you pefeed? Explain. c) In quadant 3, was thee a goup stategy that you pefeed? Explain. 6. Conside the ideas you discussed in step 5. Suggest possible stategies you could use to detemine the suface aea of each of the following 3-D objects. a) cone b) sphee c) pyamid Web Link To lean moe about Blackfoot Cossing Histoical Pak, go to and follow the links. Eagle feathe fan ove the entance to Blackfoot Exhibit Hall 2.2 Suface Aea MHR 67

3 cone a thee-dimensional object with a cicula base and a cuved lateal suface that extends fom the base to the vetex slant height adius slant height vetex base the shotest lateal distance fom the edge of the base of a cone o pyamid to its highest point lateal aea the suface that joins the two bases of a thee-dimensional object o that joins the base to the highest point lateal aea base Link the Ideas Suface Aea of a Right Cone Conside a ight cone with slant height s and base adius. Constuct a net of the cone, including the lateal aea and the base. vetex slant height, s Cicumfeence of the base = 2π s s lateal aea Fom the lateal aea of the net, Length of ac AB = 2π you can constuct a lage cicle using the vetex of the cone as the cente and the slant height of the cone, s, lateal aea as the adius. A B The lateal aea of the base of s s vetex the cone foms a secto of the lage cicle. The cicumfeence of the base of the cone, 2π, foms the length of the ac AB of the secto. The cicumfeence of the lage cicle that is fomed fom the lateal aea is 2πs. The aea of the lage cicle is πs 2. To detemine the lateal aea of the cone you can set up a popotion of coesponding atios. lateal aea of cone aea of lage cicle = cicumfeence of cone cicumfeence of lage cicle lateal aea of cone πs = _ 2π 2 2πs lateal aea of cone πs = _ 2 s Lateal aea of cone = ( _ s/ )(πs s 2 ) Lateal aea of cone = πs The lateal aea of a ight cone with adius and slant height s is πs. The base of the cone is a cicle with adius, so its aea is π 2. The total suface aea of a ight cone is the sum of the aeas of the base and the lateal suface. SA cone = π 2 + πs s base 68 MHR Chapte 2

4 Suface Aea of a Sphee The fomula fo the suface aea of a sphee is linked to the suface aea of a ight cylinde. Think of wapping a ight cylinde aound the sphee. The diamete of the sphee will be the height of the cylinde. This height will be 2. The cicumfeence of the sphee will be the cicumfeence of the cylinde. This cicumfeence will be 2π. The lateal aea of the cylinde, when flattened, foms a ectangle. The aea of the ectangle fomed by the ight cylinde is diectly elated to the suface aea of the sphee. A = (length)(width) A = (2π)(2) A = 4π 2 SA sphee = 4π 2 sphee a ound, ball-shaped object a set of points in space that ae a given distance (adius) fom a fixed point (cente) Example 1 Calculate the Suface Aea of a Right Cone A ight cone has a cicula base with diamete 27 cm and slant height 40 cm. Calculate the suface aea of the cone, to the neaest tenth of a squae centimete. 40 cm d = 27 cm Solution The suface aea of a ight cone is the sum of the aea of the single cicula base, B, and the lateal aea. SA = B + lateal aea SA = π 2 + πs SA = π(13.5) 2 + π(13.5)(40) SA = Since the diamete of the cicle is 27 cm, the adius is 13.5 cm. lateal aea base 40 cm The suface aea of the cone is appoximately cm 2. d = 27 cm You Tun Sketch a ight cone with diamete 16 cm and slant height 12 cm. What is its suface aea? 2.2 Suface Aea MHR 69

5 pyamid A thee-dimensional object with one base and the same numbe of tiangula faces as thee ae sides on the base Example 2 Calculate the Suface Aea of a Right Pyamid A ight ectangula pyamid has a ectangula base measuing 16 cm by 20 cm. The slant height of the tiangula face with the shote base is 18 cm, while the slant height of the tiangula face with the longe base is 17 cm. What is the suface aea of the pyamid? 17 cm 20 cm 18 cm 16 cm Solution As with a ight cone, the suface aea of a ight pyamid can be calculated as the sum of the aea of the base, B, and the lateal aea. 16 cm 20 cm 18 cm 17 cm SA = B + lateal aea SA = (length)(width) + 2[ 1_ 2 (length)(slant height 1 )] + 2[ 1_ 2 (width)(slant height 2 )] SA = (20)(16) + 2[0.5(20)(17)] + 2[0.5(16)(18)] SA = 948 The suface aea of the ight ectangula pyamid is 948 cm 2. The lateal aea of a ight ectangula pyamid is made up of fou tiangles. The tiangles on the opposite faces ae the same size. You Tun Sketch a ight ectangula pyamid with a squae base measuing 10 cm on each side. The slant height of each face is 8.5 cm. What is the suface aea of the pyamid? 70 MHR Chapte 2

6 Example 3 Calculate the Suface Aea of a Sphee A satellite is wapped with polyeste film to potect it duing tanspotation. How much film is equied to cove the Echo Satellite that has a cicumfeence of 95.8 m? Expess you answe to the neaest tenth of a squae mete. Solution The fomula fo the suface aea of a sphee is SA = 4π 2, whee is the adius. Calculate the adius fom the cicumfeence. C = 2π 95.8 = 2π = _ π = Substitute the adius into the fomula fo the suface aea. SA = 4π 2 SA = 4π( ) 2 SA = Appoximately m 2 of film is equied to cove the satellite. You Tun Find the suface aea of a basketball with diamete cm. Expess you answe to the neaest hundedth of a squae centimete. Example 4 Detemine a Dimension When the Suface Aea is Known The suface aea of an official 5-pin bowling ball vaies fom appoximately cm 2 to cm 2. What is the vaiation in the diamete of the bowling ball? Solution Substitute into SA = 4π 2. 4π 2 = SA 4π 2 = SA 4π 2 = π 2 = = π 2 = π = = 4π π The vaiation in the diamete of an official 5-pin bowling ball is fom 12.1 cm to 12.7 cm. How do you calculate the diamete fom the adius? You Tun To the neaest millimete, calculate the adius of a sphee with a suface aea of 1 m 2. Did You Know? Atifacts fom a game simila to bowling wee found in the tomb of an ancient Egyptian youth who died in appoximately 5200 B.C.E. Ancient Polynesians olled stones at objects fom a distance of 60 ft (18.29 m). This is the same distance as fom the foul line to the headpin in moden-day bowling. 2.2 Suface Aea MHR 71

7 Example 5 Visualize and Find Suface Aeas of Composite Objects A fam equipment manufactue in Yokton, SK, has decided to constuct a gain bin using galvanized steel. How much steel is equied to constuct the gain bin as shown? Expess you answe to the neaest hundedth 21 ft of a squae foot. Do not include ovelap of the steel sheets whee they ae fastened togethe. Solution Visualize the cylindical base with a ight cone-shaped oof. Sketch the two pats of the stuctue and add appopiate dimensions. d = 19 ft 25 ft d = 19 ft 21 ft d = 19 ft 4 ft Remembe that the bottom and top of the cylindical potion and the bottom of the conical oof ae not included. suface aea of cylindical shape = πd height SA cylinde = π(19)(21) SA cylinde = The cicumfeence of a cicle can be found using the fomula πd o 2π. To calculate the suface aea of the cone, you need to know the slant height fist. Use the Pythagoean elationship. s = s = s = s = suface aea of conical shape = πs SA cone = π(9.5)( ) SA cone = The combined suface aea is = The total suface aea of the gain bin, to the neaest hundedth of a squae foot, is ft 2. You will need a minimum of 1562 ft 2 of galvanized steel to constuct the gain bin. 72 MHR Chapte 2

8 You Tun Calculate the suface aea of the following composite object, to the neaest tenth of a squae centimete cm d = 12 cm d = 2.5 cm 6.4 cm Key Ideas The suface aea of a ight cylinde and of a ight pism can be calculated using the aea of the bases (top and bottom) plus the lateal aea. l w h SA pism = 2lw + 2lh + 2wh SA cylinde = 2(π 2 ) + 2πh The suface aea of a ight pyamid and of a ight cone can be calculated using the aea of the base plus the lateal aea. h s 2 s 1 h s l w SA pyamid = lw + 2[ 1_ 2 ls ] + 2[ 1_ 1 2 ws ] SA = 2 cone π2 + πs The suface aea of a sphee depends on the adius only. SA sphee = 4π Suface Aea MHR 73

9 Check You Undestanding Pactise Fo each of the following, expess you answes to the neaest tenth of a unit whee necessay. 1. Calculate the suface aea of each of the following. a) 1.3 m b) d = 38 in. 2.1 m 1.2 m 21 in. c) d) 8.5 in. 9 in. e) 15 cm = 3.6 cm = 4.5 cm 10 in. 8 in. 2. Sketch a ight pyamid with a squae base measuing 16 ft by 16 ft. The slant height is 12 ft. What is the suface aea of the pyamid? 3. Fo each of the following, the suface aea is given. Calculate the missing dimension. a) 34 cm SA = 6020 cm 2 20 cm b) SA = 47.4 m 2 d = 1.3 m w c) SA = 91.4 m 2 h 4. A closed box has a suface aea of 126 in. 2. The base of the box is 5 in. by 3 in. Sketch a diagam and find the height of the box. 74 MHR Chapte 2

10 5. Calculate the suface aea of this object composed of a pyamid and a sphee cm d = 8 cm 13.5 cm Apply 6. Austin is helping to build the set fo a school play. Thee ae fou cylindical pillas standing on the stage that need to be painted. Each pilla is 16 ft high and 1 ft in diamete, as shown below. Austin calculated the suface aea to the neaest hundedth of a squae foot. His wok is shown below. SA = 2(π 2 ) + πd(h) SA = 2(π)(0.5) 2 + π(1)(16) SA = 0.5π + 16π SA = 16.5π SA Thee ae fou cylindes, so SA = SA = The suface aea is ft 2. Austin thinks he made an eo in his wok. Discuss whethe Austin actually made an eo. What suface aea would you paint? 15 cm 7. One of the activities available to visitos to the Blackfoot Cossing Histoical Pak is to spend the night in a tipi. Histoically, the walls and floo of a tipi wee made of buffalo hide, but the cuent tipi is canvas with a lage cicula fabic floo. At the beginning of the season, the staff of the pak set up the tipis. While doing so, they detemine that the diamete of a tipi is 9.6 m and its slant height is 7.3 m. Assume that the tipi appoximates a cone. What is the minimum amount of canvas making up the sides of the tipi, to the neaest hundedth of a squae mete? Do not include any seam allowances. 1 ft 12 cm 16 ft 2.2 Suface Aea MHR 75

11 8. Unit Poject Compact discs ae sometimes 12 cm packaged in cylindical stacks of 100. Each CD 4.2 mm has a thickness of 1.2 mm and a diamete of 12 cm. a) The outside adius of the stoage case is 0.7 cm moe than that of the CD. The height of the case is 4.2 mm moe than that of the stack of 100 CDs. What is the suface aea of the stoage case, excluding the base, to the neaest squae centimete? b) If a ectangula CD jewel case holding a single CD is 0.5 cm wide than the CD, 2.5 cm longe than the CD, and 8 times the thickness of the CD, what is the suface aea of the jewel case? 9. A designe is woking on a line of athletic equipment. He is designing a cylindical punching bag, simila to the one shown. The designe wants to use a maximum of 1.3 m 2 of mateial to make the bag, and has detemined that the diamete of the bag should be 36 cm. Detemine the maximum possible height of the bag, to the neaest tenth of a centimete. 10. Eath has a diamete of appoximately 8000 mi. Land foms about 29% of the suface aea of Eath. Assume Eath is a sphee. Estimate the aea of land on Eath. 11. The photo shows a taditional Haida hand dum that has a diamete of 14 7_ in. and is 8 3 in. deep. What is the minimum amount of hide used to make the dum if the hide coves only the top and lateal sufaces? Expess you answe to the neaest squae inch. Taditional Haida hand dum showing twin salmon tansfoming into the next geneation. Taditional Haida hand dums ae used in ceemony, cultual events such as potlatches, and as atwok. Taditional dums should always be handled with espect following appopiate potocol. 76 MHR Chapte 2

12 12. The Muttat Consevatoy in Edmonton began with a donation fom the Muttat family. Thee ae fou geenhouses, each in the shape of a ight pyamid with a squae base. Each of the two lagest geenhouses has a base that measues 26 m on each side and has a slant height of 35.4 m. How much glass is needed fo each lage geenhouse? Expess you answe to the neaest squae mete. 13. What is the suface aea of a glass bead that has a diamete of 11 mm? Expess you answe to the neaest squae millimete. 14. Photogaphes often use a light tent to get the best lighting fo items they photogaph fo museums, catalogues, o online sales. This tent is cylindical with a conical oof. The diamete of the tent is 1 m, the height of the tent is 190 cm, and the cylindical wall of the tent is 135 cm high. What is the suface aea of the light tent, to the neaest squae mete? 135 cm 190 cm 2.2 Suface Aea MHR 77

13 Did You Know? Aound the yea 1148, the Fench played le paume, meaning the palm of the hand. This game developed into jeu de paume, eal tennis, oyal tennis, o, simply tennis. 15. Squash is a acquet spot played with a small, soft ball. The game is named fo the fact that it is vey easy to squash the ball. The ules of the game state that the ball must have a diamete of 40 mm ± 0.5 mm. This means that the actual diamete can be up to 0.5 mm moe o less than 40 mm. a) What is the minimum and maximum suface aea of a egulation squash ball? Expess you answes to the neaest squae millimete. b) Squash balls ae often packaged in cubical boxes as shown in the photo. If the box is to be sized so that the ball fits exactly in the box, find the minimum and maximum dimensions of the box. Extend Fo each of the following, expess you answes to the neaest tenth of a unit whee necessay. 2 cm 16. Gas stations often have a supply of small pape funnels that customes can use to add oil to thei vehicle engines without spilling. Each funnel is a ight cone with a small hole cut out of the top. The funnel has a slant height of 13.2 cm and the diamete of the lage opening is 9.5 cm. The diamete of the small opening is 1.5 cm. Detemine the amount of pape in the funnel. 1.5 cm 9.5 cm 13.2 cm 17. A building block is shown below. 4.6 mm 2 mm 4.6 mm 9.5 mm 31.4 mm 7.9 mm The base of the block measues 31.4 mm by 7.9 mm, and it has a height of 9.5 mm. Each ight cylinde on top of the block has a diamete of 4.6 mm and a height of 2 mm. Each hole in the block has a diamete of 4.6 mm. Detemine the suface aea of the block that needs to be painted if you do not paint the bottom of the block. 78 MHR Chapte 2

14 18. Use speadsheet softwae to help investigate how changing the adius of a sphee changes its suface aea. Ceate a speadsheet like the one shown below. a) Use speadsheet fomulas to complete the speadsheet. Depending on you softwae, you may need to type PI o PI() fo π. See you speadsheet s help featue if you need assistance. b) Compae the stetch facto fo the adius to the atio of the new suface aea to the oiginal suface aea. What patten do you notice? c) Use you patten to pedict the suface aea of the sphee if you multiply the adius by 6. Extend you speadsheet to check you answe. d) In you own wods, expess the elationship between a change in the adius of a sphee and the change in its suface aea. Ceate Connections 19. Sketch an example of a composite object with dimensions in centimetes. Explain how you can find the suface aea. Descibe how you would convet you answe to an appopiate impeial unit of aea. 20. People communicate in diffeent ways, such as oally, in witing, and even using sign language. In you own wods, explain why the suface aea of any object is expessed in squae units. Did You Know? Ameican Sign Language is the most commonly used fom of sign language in Noth Ameica. 2.2 Suface Aea MHR 79