SOLUTION: q. A closed cylindrical container has a surface area of 400cm2. Determine the dimensions for maximum volume.

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Question 577395: q. A closed cylindrical container has a surface area of 400cm2. Determine the dimensions for maximum volume.


Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
We know that

(surface area of a cylinder). Also, the volume V is



Several ways to do this. The easiest way is probably to use Lagrange multipliers, which states the maximum value of V (usually) occurs when





Equating the vector components, we have
(1)

(2)

(1) yields and (2) yields . Set these equal to each other:



Thus, the maximum volume occurs when h = 2r, or when the height equals the diameter.

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Another solution that does not use multi-variable calculus. From the very first equation we have



Plug this into your expression for V, then find dV/dr.