Question

Show that the right circular cone of least curved surface and given volume has an altitude equal to 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:
$V=\frac{1}{3}\pi r^{2}h\mathit{\Rightarrow }h\mathit{=}\frac{\mathit{3}V}{\mathrm{\pi }{r}^{\mathit{2}}}$

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 times the radius of the base.