1 6 ANALYZE OPTIMUM VOLUME SURFACE AREA EX

1. 6 ANALYZE OPTIMUM VOLUME & SURFACE AREA EX 1. MAXIMUM VOLUME FOR A GIVEN SURFACE AREA Tom has 40 m 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?

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

� We ignore negative pts � Maximum volume of cylinder is at the top of the curve: Maximum Volume of a Cylinder 20 15 �(3, Volume of Cylinder 10 5 � Interpretation: 0 0 1 2 3 -5 -10 -15 16. 5) Base Length 4 5 6 A base length of 3 gives a maximum volume for the cylinder of 16. 5 m 3 whose surface area is 40 m 2.

Maximum Volume of a Cylinder >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) Volume of Cylinder 20 15 10 5 0 -5 0 1 2 3 4 5 6 -10 -15 Base Length Maximum Volume of a Cylinder 20 Volume of Cylinder 15 10 5 0 0 0. 5 1 1. 5 2 2. 5 -5 -10 Base Length 3 3. 5 4 4. 5 5

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.

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? We use the fact that b = 2 h to reduce the number of variables Side Bottom l b = 2 h x h Front

Side � Bottom l b = 2 h x h Front

� h Side Bottom l b = 2 h x h Front TOTAL Surface Area SA = 2(AFRONT)+ABOTTOM+2(ASIDE) =2(h 2)+2 hl+2(1. 41 hl) SA = 2 h 2 + 4. 83 hl ASIDE = 1. 41 hl

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

Finding Min Surface Area of Tent 3500 3000 Surface Area of Tent 2500 2000 1500 1000 500 0 0 5 10 15 20 Height of Tent (h=9, SA=484) 25

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