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

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

Answer by richard1234(7193) (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.

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