Question

The heights of two circular cylinders of equal volume are in the ratio 1 : 2. The ratio of their radii is
(a) $1:\sqrt{2}$
(b) $\sqrt{2}:1$
(c) 1 : 2
(d) 1 : 4

Answer 1

(b) $\sqrt{2}:1$
Let the radii of the two cylinders be r and R and their heights be h and 2h, respectively.
Since the volumes of the cylinders are equal, therefore:

$\mathrm{\pi }\times r^2\times h=\mathrm{\pi }\times R^2\times 2h$
$$\begin{gathered} \Rightarrow \frac{r^2}{R^2}=\frac{2}{1} \\ \Rightarrow {\left(\frac{r}{R}\right)}^2=\frac{2}{1} \\ \Rightarrow \frac{r}{R}=\frac{\sqrt{2}}{1} \\ \Rightarrow r:R=\sqrt{2}:1 \end{gathered}$$

Hence, the ratio of their radii is $\sqrt{2}:1$.