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| Math Central | Quandaries & Queries |
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Subject: precalculus - optimization 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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