Question

A cone, a hemisphere and cylinder have equal bases. If the heights of the cone and a cylinder are equal and are same as the common radius, then find the ratio of their respective volumes.

Answer 1

Let $r$ be the common radius of the cone, hemisphere and cylinder.

Let $h$ be the common height of the cone and cylinder

Given that $r=h$

Let $V_1,V_2,V_3$ be the volumes of the cone, hemisphere and cylinder respectively.

Now $V_1:V_2:V_3=\frac{1}{3}\pi r^{2}h:\frac{2}{3}\pi r^3:\pi r^{2}h$

$\Rightarrow \frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3:\pi r^3$ (given $r=h$)

$\Rightarrow V_1:V_2:V_3=\frac{1}{3}:\frac{2}{3}:1$

Hence the required ratio is $1:2:3$