Two circular cylinders of equal volume have their heights in the ratio 1 : 2.

Question:

Two circular cylinders of equal volume have their heights 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

Solution:

(b) $\sqrt{2}: 1$

Suppose that the heights of two cylinders are and 2h whose radii are r and R, respectively.
Since the volumes of the cylinders are equal, we have:

$\pi r^{2} h=\pi R^{2} \times 2 h$

$\Rightarrow \frac{r^{2}}{R^{2}}=\frac{2}{1}$

$\Rightarrow\left(\frac{r}{R}\right)=\sqrt{\frac{2}{1}}$

$\Rightarrow r: R=\sqrt{2}: 1$