SOLUTION: Find the dimensions (radius and height) that give the minimum cost for producing cylindrically shaped aluminum cans with lids and a volume of 16pi in^3. How much would it cost to p

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Question 1118294: Find the dimensions (radius and height) that give the minimum cost for producing cylindrically shaped aluminum cans with lids and a volume of 16pi in^3. How much would it cost to produce 100,000 cans if aluminum price is 0.03 cents/in^2?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Look for minimum AREA for the minimum cost.

h, height
r, radius
A, area

system%28h%2Api%2Ar%5E2=16pi%2CA=2%2Api%2Ar%5E2%2Bh%2A2pi%2Ar%29
-
hr%5E2=16
h=16%2Fr%5E2
-
A=2pi%2Ar%5E2%2B%2816%2Fr%5E2%29%282pi%2Ar%29
A=2pi%2Ar%5E2%2B32pi%2Fr------------------(technology is showing minimum at about r=2)
dA%2Fdr=4pi%2Ar%2B%2832pi%29%28-1%29%28r%5E%28-2%29%29

dA%2Fdr=4pi%2Ar-32pi%2Fr%5E2
common_denominator_r%5E2
dA%2Fdr=%284pir%5E3-32pi%29%2Fr%5E2
Need the NUMERATOR to be 0.
4pi%2Ar%5E3-32pi=0
4pi%2Ar%5E3=32pi
r%5E3=8
highlight%28r=2%29
.
With this r value found for minimum A, the value for h, minimum A, can be evaluated, and also the corresponding cost.