SOLUTION: The total surface area of a can is 384pi square inches. Find the dimensions of the can if the volume is a maximum.

Question 694379: The total surface area of a can is 384pi square inches. Find the dimensions of the can if the volume is a maximum.
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
V = pi*r^2*L
Surface area, S = 2*pi*r*L + 2*pi*r^2 = 384*pi
First you would solve for L in terms of r.
Doing the algebra you will get L = 192/r - r
The volume will be maximum where dV/dr = 0
Taking the derivative and carrying out the algebra you will get the expression
3r^2 = 192 -> r = 8
Therefore L = 192/8 - 8 = 16
Ans: length 16 in., radius 8 in.
We can see that the answer is correct by plotting V as a function of r
V=pi*r^2*(192/r - r); r along the x-axis:

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