[Math] optimize area of cylinder with no givens

calculusoptimization

I have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads:

A metal can is in the form of a cylinder. It has a bottom but no top. The surface area is a constant S square meters. Find the dimensions of the can which maximize the volume.

For the surface area of a cylinder, I know the equation is $2\pi rh + 2\pi r^2$. Since we have no top of the can, the formula changes to $2\pi rh + \pi r^2$. The volume of a cylinder is $V=\pi r^2h$.

In all of the other problems on the sheet we've basically solved for one variable like r or h using a given surface area or volume. This problem, however says the surface area is $S$ instead of actually giving a numeric value.

Is there any way to solve this problem?

Best Answer

Use the equation

$$ S= 2\pi r h + \pi r^2 $$

to get $h$ as

$$ h=\frac{S-\pi r^2 }{2\pi r } $$

then substitute in the volume equation V for $h$ to get an equation in $r$

$$ V(r) = \pi r^2 \left(\frac{S-\pi r^2 }{2\pi r } \right) $$

and then use derivative techniques to maximize.

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