Problem Statement
Show that the volume of the greatest cylinder that can be inscribed in a sphere of radius $R$ is $\frac{4\pi R^3}{3\sqrt{3}}$.
Verified Solution & Marking Scheme
Express Volume in Terms of Height h
Let cylinder height be $h$ and radius be $r$. Then $r^2 + (h/2)^2 = R^2 \implies r^2 = R^2 - \frac{h^2}{4}$. Volume $V = \pi r^2 h = \pi h \left(R^2 - \frac{h^2}{4}\right) = \pi \left(R^2 h - \frac{h^3}{4}\right)$.
Find Stationary Point
$\frac{dV}{dh} = \pi \left(R^2 - \frac{3h^2}{4}\right) = 0 \implies h^2 = \frac{4R^2}{3} \implies h = \frac{2R}{\sqrt{3}}$
$\frac{d^2V}{dh^2} = -\frac{3\pi h}{2} < 0$, verifying local maximum.
Compute Maximum Volume
$V_{\max} = \pi \left( R^2 \cdot \frac{2R}{\sqrt{3}} - \frac{1}{4}\left(\frac{8R^3}{3\sqrt{3}}\right) \right) = \pi R^3 \left( \frac{2}{\sqrt{3}} - \frac{2}{3\sqrt{3}} \right) = \frac{4\pi R^3}{3\sqrt{3}}$