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EX 1. MAXIMUM VOLUME FOR A GIVEN SURFACE AREA Tom has 40m 2 of plastic sheeting to build a greenhouse in the shape of a square-based prism. What are the dimensions that will provide the maximum volume, assuming she must cover all six sides with sheeting?

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Insert ‘h’ into Volume equation This equation will allow us to determine MAX volume

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We ignore negative pts Maximum volume of cylinder is at the top of the curve: (3, 16.5) Interpretation: A base length of 3 gives a maximum volume for the cylinder of 16.5m 3 whose surface area is 40m 2.

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>What could go wrong with our answer? >We could be missing the maximum point by a bit >Solution? Plot more points >With new graph, max point is now (2.5,17.1875) instead of (3, 16)

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An outdoor sporting goods manufacturer is designing a new tent in the shape of an isosceles right triangular prism. To maintain the shape of this prism, the base of the triangular face must always be double its height.

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To fit 5 people comfortably with gear, the volume inside the tent needs to be 600 ft 3. What dimensions will give this tent a minimum surface area? Front SideBottom b = 2h h l We use the fact that b = 2h to reduce the number of variables x

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Front SideBottom b = 2h h l x

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Front SideBottom b = 2h h l x h h x A SIDE = 1.41hl TOTAL Surface Area SA = 2(A FRONT )+A BOTTOM +2(A SIDE ) =2(h 2 )+2hl+2(1.41hl) SA = 2h 2 + 4.83hl

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Now need to substitute this expression for ‘l’ into the equation for Minimum Surface Area

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(h=9, SA=484)

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p. 60 #2, 6, 7, 9, 12

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