Question

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.

Answer 1

Let x be the radius of the base and h be the height of the cone which is inscribed in a sphere of radius r.

In the figure, AD = h and CD = x = BD Since, ΔABD and ΔBDE are similar,

Let V be the volume of the cone.

For maximum volume, dV/dh = 0

∴ V is maximum when h = 4r/3

Hence, the altitude (i.e. height) of the right circular cone of maximum volume = 4r/3.