Question

A cone, a hemisphere and a cylinder stand on the same base and have equal height. Find the ratio of their volumes.

Answer 1

Let, radius of bases be '$r$'

and their height be '$h$'

For hemisphere,

$radius=height⟹r=h......\left(1\right)$

Now,

volume of cone, $v_1=\frac{1}{3}\pi r^{2}h$

volume of hemisphere, $v_2=\frac{2}{3}\pi r^3$

volume of cylinder, $v_3=\pi r^{2}h$

The ratio, $v_1:v_2:v_3=\frac{1}{3}\pi r^{2}h:\frac{2}{3}\pi r^3:\pi r^{2}h$

$=\frac{h}{3}:\frac{2r}{3}:h$

$=\frac{h}{3}:\frac{2h}{3}:h$ [From $\left(1\right)$]

$=\frac{1}{3}:\frac{2}{3}:1$

$=1:2:3$

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