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:
Let r be the radius of the base.
and h be the height.
Here, h = r.
Now,
The ratio of their volumes will be
Volume of cone : volume of hemisphere : volume of a cylinder
$\frac{1}{3} \pi r^{2} h: \frac{2}{3} \pi r^{3}: \pi r^{2} h$
$V_{1}: V_{2}: V_{3}=\frac{1}{3} \pi r^{3}: \frac{2}{3} \pi r^{3}: \pi r^{3}$
Hence, $\quad V_{1}: V_{2}: V_{3}=1: 2: 3$