A solid right circular cylinder has a hemisphere hollowed out from each end. Given that the total surface area is 64pi cm sq, find the radius of the cylinder when the volume is maximum.
Surface area
$\displaystyle 2\pi r^2+2\pi rh-6\pi r^2=64\pi $ simplifies to ..
$\displaystyle h=\frac{32+2r^2}{r}$
Volume ..
$\displaystyle V=\pi r^2h-\frac{4}{3}\pi r^3$
Substitute h into this equation . After that , calculate $\displaystyle \frac{dV}{dr}$ then have it equals to 0 to find r .