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.

Let r and h be the radius and height of the cone respectively.

Then, the volume (V) of the cone is given by:

V=

The surface area (S) of the cone

S = πrl

=

Then,

Now, if

So, when then > 0

Then, by second derivative test, the surface area of the cone is the least when .

So when then h =

Therefore, 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.