The volume of the greatest cyliner which can be inscribed in a cone of height 30 cm and semivertical angle ${30}^0$  is

From geometry, we have

$\frac{r}{30\tan {30}^{\circ }}=\frac{30-h}{30}$

or  $h=30-\sqrt{3}r$

Now, the volume of cyclinder,

$V=\pi r^{2}h=\pi r^2(30-\sqrt{3}r)$

Now, let $\frac{dV}{dr}=0 \text{or }\pi (60r-3\sqrt{3}{r}^2)=0 \text{or }r=\frac{20}{\sqrt{3}}$

Hence,  $V_{max}=\pi {\left(\frac{20}{\sqrt{3}}\right)}^2(30-\sqrt{3}\frac{20}{\sqrt{3}})=\pi \frac{400}{3}\times 10$

$=\frac{4000\pi }{3}$