Question

A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.

Answer 1

Let r be the radius of hemisphere & cone and h be the height of the cone.
$\therefore Volumeofcone=\frac{1}{3}\pi r^{2}h$
and volume of hemisphere $=\frac{2}{3}\pi r^3$
$\because$ Their volumes are equal Now ratio in their heights
$=\frac{1}{3}\pi r^{2}h=\frac{2}{3}\pi r^3\Rightarrow r^{2}h=2r^3\Rightarrow h=2r\Rightarrow \frac{h}{r}=\frac{2}{1}$
$\therefore$ Ratio in their heights = 2 : 1