The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

$\mathrm{h}=2 \mathrm{rsin} \theta$

$\mathrm{a}=2 \mathrm{rcos} \theta$

$\mathrm{v}=\pi(\mathrm{r} \cos \theta)^{2}(2 \mathrm{r} \sin \theta)$

$\mathrm{v}=2 \pi \mathrm{r}^{3} \cos ^{2} \theta \sin \theta$

$\frac{\mathrm{dv}}{\mathrm{d} \theta}=\pi \mathrm{r}^{3}\left(-2 \cos \theta \sin ^{2} \theta+\cos ^{3} \theta\right)=0$

or $\tan \theta=\frac{1}{\sqrt{2}}$

$\because h=2 \times 3 \times \frac{1}{\sqrt{3}}$

$=2 \sqrt{3}$