TITLE: Height/Radius ratio for maximum volume cylinder of given surface area
QUESTION [5 upvotes]: I am a bit confused by this problem I have encountered:
A right circular cylindrical container with a closed top is to be constructed
with a fixed surface area. Find the ratio of the height to the
radius which will maximize the volume.
I know the volume to be $ \pi{r}^2h$, but I don't see what equation I should be solving for. How can I solve for the ratio?
Thanks

REPLY [3 votes]: The surface area $S=2\pi r^2+2\pi rh$ is constant, 
so $\displaystyle\frac{dS}{dr}=4\pi r+2\pi r\frac{dh}{dr}+2\pi h=0\implies 2r + r\frac{dh}{dr}+h=0$.
When the volume $V=\pi r^2h$ is a maximum, 
$\;\;\;\displaystyle\frac{dV}{dr}=\pi r^2\frac{dh}{dr}+2\pi rh=0\implies r\frac{dh}{dr}=-2h$.
Therefore the volume is a maximum when $2r-2h+h=0,\;$ so $h=2r$ and $\displaystyle\frac{h}{r}=2$.