If the ratio of volumes of two spheres is 1 : 8, then the ratio of their surface areas is :

1. 1 : 2

2. 1 : 4

3. 1 : 8

4. 1 : 16

Let radius of two spheres be r and R.

Given,

Ratio of volume of the two spheres is 1 : 8.

Volume of sphere = $\dfrac{4}{3}$ π.(radius)³

$\therefore \dfrac{\text{Volume of Sphere 1}}{\text{Volume of Sphere 2}} = \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{\dfrac{4}{3} \times π \times \text{r}^3}{\dfrac{4}{3} \times π \times \text{R}^3} = \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{\text{r}^3}{\text{R}^3} = \dfrac{1}{8} \\[1em] \Rightarrow \Big(\dfrac{\text{r}}{\text{R}}\Big)^3 = \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{\text{r}}{\text{R}} = \sqrt[3]{\Big(\dfrac{1}{8}\Big)} \\[1em] \Rightarrow \dfrac{\text{r}}{\text{R}} = \dfrac{1}{2}.$

Surface area of sphere = 4π.(radius)²

$\therefore \dfrac{\text{Surface area of Sphere 1}}{\text{Surface area of Sphere 2}} = \dfrac{1}{8} \\[1em] = \dfrac{4 \times π \times \text{r}^2}{4 \times π \times \text{R}^2} \\[1em] = \Big(\dfrac{\text{r}}{\text{R}}\Big)^2 \\[1em] = \Big(\dfrac{1}{2}\Big)^2 \\[1em] = \dfrac{1}{4} \\[1em] = 1 : 4.$

Hence, option 2 is the correct option.