Question:
A cone, a hemisphere, and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1:2:3.
Solution:
Given that
A cone, a hemisphere and a cylinder stand on one equal bases and have the same weight
We know that
$\mathrm{V}_{\text {cone }}: \mathrm{V}_{\text {hemisphere }}: \mathrm{V}_{\text {cylinder }}$
$1 / 3 \pi r^{2} h: 2 / 3 \pi r^{3}: \pi r^{2} h$
Multiplying by 3
$\pi r^{2} h: 2 \pi r^{3}: 3 \pi r^{2} h$
$\pi r^{3}: 2 \pi r^{3}: 3 \pi r^{3}\left(\therefore r=h\right.$ and $\left.r^{2} h=r^{3}\right)$
1:2:3 Therefore the ratio is 1: 2: 3.