Question

Show that the right circular cone of least curved surface and given volume has an altitude equal to√2  time the radius of the base.

Answer 1

Let r and h be the radius and the height (altitude) of the cone respectively.
Then, the volume (V) of the cone is given as:

The surface area (S) of the cone is given by, S = πrl (where l is the slant height)

Thus, it can be easily verified that when  

∴ By second derivative test, the surface area of the cone is the least when 

Hence, for a given volume, the right circular cone of the least curved surface has an altitude equal to √2  times the radius of the base.