Hi folks,

A cylinder of volume V is to be cut from a solid sphere of radius R. Prove that the maximum value of V is $ \frac{4 \pi R^3}{3 \sqrt{3}}$

Volume of the cylinder V = $\pi r^2 h$

From the attached drawing $R^2 = r^2 + \frac{h^2}{4}$

making h the subject of this equation, we get $h = 2 \sqrt{R^2 - r^2}$ and substitute back

V = $2 \pi r^3 (\frac{R^2}{r^2} - 1)^\frac{1}{2}$

The idea is to differentiate V with respect to r, set to 0 (for a max) and get the terms in r to drop out, but the expression is quite a handful and I suspect I am on the wrong track.

Any advice?