Relation between the height $h$ and the radius $r$ subject to given conditions

Question

What is the relation between the height $h$ and the radius $r$ of a right circular cylinder of fixed volume $V$ and minimal total surface area?

Answer 1

The volume $V=\pi r^2h$, the surface area $A=2\pi r^2+2\pi rh$. $V$ is fixed, so $A=2\pi r^2+2V/r$.

Differentiating we get $4\pi r-2V/r^2$. That is 0 for $r=\left(\frac{V}{2\pi}\right)^{1/3}$. It is negative for smaller $r$ and positive for larger $r$, so a minimum.

That gives $h=2r$. So the required cylinder has its height equal to its diameter.