SOLUTION: calculus. A cylindrical container is to hold 20π cm3. The bottom is made of a material that costs $0.80 per cm2, and the top is left open (no material needed). The material fo

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Question 858609: calculus. A cylindrical container is to hold 20π cm3. The bottom is made of a material that costs $0.80 per cm2, and the top is left open (no material needed). The material for the curved side costs $0.32 per cm2.
Find the height in centimeters of the most economical container.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Let h = height, r = radius.
Let A = surface area
Let v=volume

A=2%2Api%2Ar%5E2%2Ah%2Bpi%2Ar%5E2.
v=h%2Api%2Ar%5E2=20pi.
Solve this for h:
h=20%2F%28r%5E2%29, and substituting into the A equation,
A=2%2Api%2Ar%2820%2F%28r%5E2%29%29%2Bpi%2Ar%5E2
A=40%2Api%2Fr%2Bpi%2Ar%5E2,which you will use for applying the costs of the two different parts.

Now, applying the cost to those two area parts, cost as a function of r, C(r) becomes:
C%28r%29=%280.32%2940%2Api%2Fr%2B%280.80%29pi%2Ar%5E2
highlight_green%28C%28r%29=12.8%2Api%2Fr%2B%280.80%29pi%2Ar%5E2%29.
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You want to minimize the cost. Find the derivative of C with regard to r, set equal to zero, and solve this for r. You then use this to find your value of h.

You might also want to check your result using a graphing calculator for the cost equation without use of Calculus.