Find maximum volume of sphere inscribed inside a cone of circular base

Question

A sphere is inscribed in a right circular cone with volume of $10$ $cm^3$. Find the maximum volume of the sphere.

Answer 1

From the cross-section diagram, the volumes of the cone and the sphere are respectively, $$V_c=\frac13 \pi R^3\tan2\theta,\>\>\>\>\>V_s = \frac{4}3\pi R^3\tan^3\theta$$

Eliminate $R$,

$$V_s = \frac{4\tan^3\theta}{\tan2\theta}V_c=2\tan^2\theta(1-\tan^2\theta)V_c$$

Set $V_s'(\theta) = 0$ to find the maximum value at $\tan\theta = \frac1{\sqrt2}$. Then, plug it into above expression to obtain the maximum volume of the sphere,

$$V_s^{max} = \frac12 V_c$$

which happens to be half volume of the cone.