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 Subject: precalculus - optimization Name: ashley Who are you: Student A cylindrical container costs $2.00 per square foot for the sides and$3.00 a square foot for the top and bottom. The container must hold 100 cubic feet of material. What are the dimensions of the most economical container.

Hello Ashley.

The dimensions of a cylinder are the height, h, and the radius r. So the cost of the cylinder is

C(h,r) = 2(π r2)3+(2 π r h)2 = 6 π r2 + 4 π r h

The volume of a cylinder is V = π r2 h, so if V=100, then

100 = π r2h
h = 100/(π r2)

If we substitute this expression for h into the cost equation, then:

C(r) = 6 π r2 + 4 π r(100/(π r2))
C(r) = 6 π r2 + 400/r

Now you have a single equation with two variables. Can you do the final step and find the value of r that gives the smallest C(r)?

Hope this helps,
Stephen La Rocque.

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