SOLUTION: A manufacturer of food-storage containers makes a cylindrical bin with a volume of 1000cm. What dimensions (height and radius) will minimize the material needed to produce each bin

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Question 940367: A manufacturer of food-storage containers makes a cylindrical bin with a volume of 1000cm. What dimensions (height and radius) will minimize the material needed to produce each bin, that is minimize the surface area?
1000 = Pai r^2 h
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Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The volume of the can is,
V=pi%2AR%5E2%2AH=1000
The surface area of the can is,
A=2pi%2AR%5E2%2B2pi%2AR%2AH
From the volume equation,
H=1000%2F%28piR%5E2%29
Substitute into the surface area equation,
A=2pi%2AR%5E2%2B2pi%2AR%281000%2Fpi%2AR%5E2%29
A=2pi%2AR%5E2%2B2000%2FR%29
Take the derivative of area with respect to R and set it equal to zero.
dA%2FdR=4pi%2AR-2000%2FR%5E2=0
4pi%2AR=2000%2FR%5E2
R%5E3=500%2Fpi
R=%28500%2Fpi%29%5E%281%2F3%29
SO then,
H=1000%2F%28pi%2AR%5E2%29
H=1000%2F%28pi%2A%28500%2Fpi%29%5E%282%2F3%29%29
H=%281000%2F500%5E%282%2F3%29%29%281%2Fpi%5E%281%2F3%29%29