Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. The ratio of their radii is :

1. 1 : $\sqrt{2}$

2. $\sqrt{2}$ : 1

3. 1 : 2

4. 1 : 4

Let radius and heights of two cylinders be r, h and R, H.

Given,

$\dfrac{\text{h}}{\text{H}} = \dfrac{1}{2}$

Volume of cylinder 1 = v

Volume of cylinder 2 = V

⇒ v = V

$\Rightarrow π\text{r}^2\text{h} = π\text{R}^2\text{H} \\[1em] \text{Divide by π on both sides, we get:} \\[1em] \Rightarrow \text{r}^2 = \text{R}^2 \times \dfrac{\text{H}}{\text{h}} \\[1em] \Rightarrow \dfrac{\text{r}^2}{\text{R}^2} = \dfrac{2}{1} \\[1em] \Rightarrow \Big(\dfrac{\text{r}}{\text{R}}\Big)^2 = \dfrac{2}{1} \\[1em] \Rightarrow \dfrac{\text{r}}{\text{R}} = \sqrt{\dfrac{2}{1}} \\[1em] \Rightarrow \dfrac{\text{r}}{\text{R}} = \dfrac{\sqrt{2}}{1} \\[1em]$

∴ r : R = $\sqrt{2}$ : 1

Hence, option 2 is the correct option.