Question

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.

Answer 1

Let r and h be the radius and height of the cone respectively inscribed in a sphere of radius R.

Let V be the volume of the cone.

Then,

Height of the cone is given by,

h = R + AB

∴ By second derivative test, the volume of the cone is the maximum when