1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis potograp. Sometimes owever, it is useful to develop algebraic and grapical models tat can be analysed wit greater precision. Example 1 Maximum Volume for a Given Surface Area Tillie as 40 m 2 of plastic seeting to build a greenouse in te sape of a square-based prism. Wat are te dimensions tat will provide te maximum volume, assuming se must cover all six sides wit seeting? Solution Te greenouse is to be in te sape of a square-based prism. Develop algebraic models to describe te volume and surface area of te greenouse. 54 MHR Capter 1 01_FFCM12_CH1.indd 54 3/6/09 12:17:38 PM
Volume V = Base area Heigt V = s 2 Surface Area S.A. = 2 Base area + 4 Side area S.A. = 2s 2 + 4s Rearrange tis formula to isolate. S.A. = 2s 2 + 4s S.A. - 2s 2 = 4s _ S.A. - 2s 2 = _ 4s 4s 4s S.A. - 2s2 = _ 4s Substitute tis expression into te volume equation. V = s 2 ( _ S.A. - 2s2 4s ) Substitute S.A. = 40 m 3, and simplify te equation. V = s ( 2 _ 40-2s2 4s ) V = _ 40s 4 - _ 2s3 4 V = 10s - 0.5s 3 Determine te maximum volume. Determine te base side lengt tat will give te maximum volume. Use a graping calculator. Use reasoning and systematic trial to coose appropriate window settings. Determine te coordinates of te maximum point on tis grap. Press 2nd [CALC]. Select 4:maximum. Follow te prompts, pressing after eac step. 1.6 Analyse Optimum Volume and Surface Area MHR 55 01_FFCM12_CH1.indd 55 3/6/09 12:17:39 PM
Te maximum volume of te greenouse is approximately 17.2 m 3, wic occurs wen te base side lengt is approximately 2.58 m. Use s = 2.58 to determine te eigt of te greenouse. V = s 2 17.2 = 2.58 2 17.2 6.66 _ 17.2 6.66 = _ 6.66 6.66 2.58 Te dimensions of te greenouse tat give te maximum volume of 17.2 m 3 are 2.58 m by 2.58 m by 2.58 m. Te maximum volume of a square-based prism occurs wen te side lengt of te square base is equal to te eigt. Example 2 Minimum Surface Area for a Given Volume An outdoor sporting goods manufacturer is designing a new tent in te sape of an isosceles rigt triangular prism. eigt, lengt, l base, b To maintain te sape of tis prism, te base of te triangular face must always be double its eigt. b = 2 To fit five people comfortably wit gear, te volume inside te tent needs to be 600 ft 3. Wat are te dimensions of a tent wit a volume of 600 ft 3 and a minimum surface area? 56 MHR Capter 1 01_FFCM12_CH1.indd 56
Solution Draw a net for te triangular prism. side bottom l x b = 2 front Develop simplified algebraic expressions for te area of eac face. Use te relationsip b = 2 to reduce te total number of variables. Front and Back Faces Te front and back faces are rigt isosceles triangles. A = 1 _ 2 b = 1 _ 2 (2) = 1 _ 2 (2) = Substitute b = 2. Simplify te expression. Floor Te floor of te tent is a rectangle wit dimensions b l. A = bl = (2)l Substitute b = 2. = 2l Sides Te sides are rectangles wit lengt l. Te widt is equal to te equal sides on te triangular faces. side x b = 2 front Te triangular face can be broken into two small triangles, eac wit legs units long and ypotenuse x units. Te two legs of tis triangle are bot equal to. 2 2 = x 1.6 Analyse Optimum Volume and Surface Area MHR 57 01_FFCM12_CH1.indd 57
Use te Pytagorean teorem. x 2 = + x 2 = 2 2 x 2 = 2 x = 2 x 1.4142 Collect like terms. Take te square root of bot sides. Use tis value to determine an expression for te area of one side of te tent. A 1.4142l side l 1.4142 Total Surface Area Add te areas of te faces to develop an expression for te surface area of te tent. S.A. = 2(A Front ) + A Bottom + 2(A Side ) 2( ) + 2l + 2(1.4142l) = 2 + 4.8284l Collect like terms and simplify. Te volume needs to be 600 ft 3. Write te volume in terms of and l ten simplify te expression for surface area. V = Base area Heigt = ( 1 _ 2 b ) = [_ 1 2 (2) ] V = l 600 = l _ 600 = _ 2 l _ l = 600 _ Substitute l = 600 of te eigt of te tent. Substitute b = 2 and simplify. Substitute V = 600. Divide bot sides by. to develop an expression of surface area in terms S.A. = 2 + 4.8284l = 2 + 48284 ( _ 600) = 2 + 48284 ( _ 600) = 2 + _ 2897 Determine te minimum surface area. Te equation S.A. = 2 + _ 2897 gives te surface area for any isosceles rigt triangular prism wit volume 600 ft 3. 58 MHR Capter 1 01_FFCM12_CH1.indd 58
Grap te equation to determine te eigt tat minimizes te surface area. Use a graping calculator. Determine te coordinates of te minimum point on tis grap. Press 2nd [CALC]. Select 3:minimum. Follow te prompts, pressing ENTER after eac step. Te minimum surface area of te tent is 484 ft 2, wic occurs wen te eigt is approximately 9 ft. Determine te dimensions of te tent. Use 9 to determine te oter dimensions of te tent. b = 2 Te base is twice te eigt. = 2(9) = 18 Te base of te triangular face is approximately 18 ft. _ l = 600 _ = 600 (9) 2 7.4 Te lengt of te tent is approximately 7.4 ft. Recall tis relationsip between te lengt and eigt of te tent. Te dimensions of an isosceles rigt triangular based prism saped tent, aving a volume of 600 ft 2, and minimum surface area are sown. = 9 ft l = 7.4 ft b = 18 ft 1.6 Analyse Optimum Volume and Surface Area MHR 59 01_FFCM12_CH1.indd 59
Key Concepts Te minimum surface area for a given volume of a square-based rectangular prism occurs wen te eigt is equal to te side lengt of te base. Te minimum surface area for a given volume of a cylinder occurs wen te eigt is equal to te diameter. Tere are a number of manipulative, tecnological, and algebraic tools and strategies tat are useful wen optimizing volume or surface area of prisms and cylinders. Discuss te Concepts D1. Two square-based prism saped boxes bot ave a volume of 48 cm 3. Can it be concluded tat tese boxes ave te same surface area? If yes, explain wy. If no, provide a counter-example. D2. Suppose you use two identical standard seets of paper. You roll one lengtwise to form a cylinder and you roll te oter widtwise to form a cylinder. a) Suppose tat te cylinders are popcorn containers (wit a bottom). Wic do you tink would old more popcorn? Explain wy you tink so. b) Test your conjecture. D3. Do te containers in question D2 ave te same surface area? Discuss wit a partner. Practise A 1. Eac sed as te same volume. Witout measuring, order tese seds from minimum to maximum surface area. Do not include te bottom of te seds. Explain your reasoning. A B C D 60 MHR Capter 1 01_FFCM12_CH1.indd 60
2. Eac cylindrical container as te same surface area. Apply B E F G H Witout measuring, order tese containers from maximum to minimum volume. Explain your reasoning. 3. A candy store, in te sape of a square-based prism is to ave a volume of 1000 m 3. Determine te dimensions of te store wit te minimum surface area. 4. Refer to question 3. How would your answer cange if te minimum surface area did not include te bottom of te building? Explain. 5. A slipper manufacturer is releasing a new product called Rocco and Biff Warm n Fuzzies. Te slippers are to be sipped in boxes in te sape of square-based prisms, 50 pairs to a box. Te slippers can be arranged different ways, but eac slipper requires 11 000 cm 3 of space. a) Determine te volume of eac sipping box. b) Wat are te dimensions of te box wit a minimum surface area? c) Sketc te box and label its dimensions. 6. Carlie s Cick Peas come in cans 8 cm as sown. a) Determine te volume of one can. b) Could Carlie save money on packaging materials by altering te design of is can? Explain. c) Determine te maximum amount Carlie can save on materials witout reducing te volume of te container. Express your answer as a percent. 18 cm 1.6 Analyse Optimum Volume and Surface Area MHR 61 01_FFCM12_CH1.indd 61
7. A soup can is to old 600 ml of soup. Determine te dimensions of te can wit minimum surface area. Reasoning and Proving Representing Selecting Tools 8. A producer of nutrition bars is designing a bar tat will just fit inside a package in te sape of an equilateral triangular-based prism. Problem Solving Connecting Reflecting Communicating a) Determine te dimensions of a 350-mL bar tat requires a minimum amount of packaging material. b) Describe te tools and strategies you used to solve tis problem, and any assumptions you made. Acievement Ceck 9. Jessie as 24 m 2 of wood wit wic to build a tree ouse. Jessie plans to build te tree ouse in te sape of a square-based prism. Wat is te maximum volume se can enclose? 10. Refer to question 9. How muc additional volume can be added to te tree ouse in eac case? i) Jessie leaves te roof open. ii) Jessie leaves te roof and back wall (facing te tree trunk) open. Explain your solution in eac case, including any assumptions tat you made. 62 MHR Capter 1 01_FFCM12_CH1.indd 62 3/6/09 12:17:41 PM
Extend C 11. Find an object in your classroom or at ome tat uses geometrically saped packaging, suc as a tissue box or a can of food. a) Take measurements and determine te volume of te object. b) Calculate te surface area of te object. c) Is te object packaged optimally? Explain. d) Determine te dimensions of a package aving te same sape and volume but wit a minimum surface area. e) Suggest reasons wy objects in stores are not all packaged optimally. 12. Wen designing an apartment or office building, te sape of te rectangular prism wit minimum surface area is a cube. Many buildings in towns and small cities are close to cube-saped. However, in te downtown areas of many large cities, tis sape is less common. Wat type of sape is more common in cities? Wy do you tink tis is so? 13. Can you tink of a situation in wic a retailer may not wis to optimize te volume of a package or container for a given surface area? Provide an example and explain wy tey migt not wis to do so. 14. A tent is in te sape of a triangular prism wen constructed. a) Use Te Geometer s Sketcpad to sow wy te base of te triangular face of te tent must always be double te eigt in order to preserve te sape of te face. b) Use algebraic and geometric reasoning to prove tis fact. 15. For a given surface area, wic type of container can old te greatest volume: a square-based prism or a cylinder? How did you determine te answer? 16. A bar of soap is in te sape of a rectangular-based prism, wit dimensions 1 cm by 3 cm by 6 cm. Determine te dimensions of a cylindrical container tat will be just large enoug to old 12 bars of soap. Describe ow te bars sould be stacked inside te container. 17. A cylindrical can of speciality candies as been designed to minimize te surface area. It is packaged in a rectangular box tat just fits te can. Does te box also ave te minimum surface area for te volume enclosed? Justify your answer. 1.6 Analyse Optimum Volume and Surface Area MHR 63 01_FFCM12_CH1.indd 63 3/6/09 12:17:41 PM