Question

Find the dimensions of the right circular cone of least volume that can be circumscribed about a sphere of radius a. Hint: x+2a=altitude of cone 

Answer 1

In Triangle AOB,

$AO^2+AB^2=OB^2$

$(h-a)^2+r^2=a^2$

$\implies r^2 = a^2-(h-a)^2$

Volume of the Cone, $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi [ a^2-(h-a)^2 ]h$

For maximum and minimum,

$\frac{dV}{dh}=0$

Then,

$\frac{dV}{dh}= \frac{1}{3}\pi[ -2(h-a)h+a^2-(h-a)^2 ] =0$

$\implies h = \frac{4}{3}a$

Then, $r =\frac{2\sqrt{2}}{3}a$

Then, Volume will be, $V = \frac{32}{81}\pi a^3$