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

Answer 1

(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:

$$\begin{gathered} \frac{{\displaystyle \frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{\mathit{2}}h}{{\displaystyle \frac{2}{3}}\mathrm{\pi }{r}^3}}}{\mathrm{\pi }{r}^{2}h} \\ =\frac{{\displaystyle \frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{\mathit{3}}}{{\displaystyle \frac{2}{3}}\mathrm{\pi }{r}^3}}}{\mathrm{\pi }{r}^3} \\ =\frac{{\displaystyle \frac{1}{2}}}{3}=1:2:3 \end{gathered}$$