Question:
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is
(a) 1 : 2 : 3
(b) 2 : 1 : 3
(c) 2 : 3 : 1
(d) 3 : 2 : 1
Solution:
(a) 1 : 2 : 3
The cone, hemisphere and the cylinder stand on equal bases and have the same height.
We know that radius and height of a hemisphere are the same.
Hence, the height of the cone and the cylinder will be the common radius.
i.e., r = h
Ratio of the volumes of the cone, hemisphere and the cylinder:.
$\frac{\frac{\frac{1}{3} \pi r^{2} h}{\frac{2}{3} \pi r^{3}}}{\pi r^{2} h}$
$=\frac{\frac{\frac{1}{3} \pi r^{3}}{\frac{2}{3} \pi r^{3}}}{\pi r^{3}}$
$=\frac{\frac{1}{2}}{3}=1: 2: 3$