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
to get $h$ as
then substitute in the volume equation V for $h$ to get an equation in $r$
and then use derivative techniques to maximize.