The ratio between the volumes of two spherical solids is 27 : 8. The ratio between their curved surface areas is :

1. 27 : 8

2. 8 : 27

3. 3 : 2

4. 9 : 4

Let radius of two spherical solids be R and r.

Given,

The ratio between the volumes of two spherical solids is 27 : 8.

$\Rightarrow \dfrac{\text{Vol. of 1st spherical solid}}{\text{Vol. of 2nd spherical solid}} = \dfrac{27}{8} \\[1em] \Rightarrow \dfrac{\dfrac{4}{3}πR^3}{\dfrac{4}{3}πr^3} = \dfrac{27}{8} \\[1em] \Rightarrow \dfrac{R^3}{r^3} = \dfrac{27}{8} \\[1em] \Rightarrow \Big(\dfrac{R}{r}\Big)^3 = \Big(\dfrac{3}{2}\Big)^3 \\[1em] \Rightarrow \dfrac{R}{r} = \dfrac{3}{2} \text{ ………(1)}$

The ratio between the curved surface area of two spherical solids :

$\Rightarrow \dfrac{\text{CSA of 1st spherical solid}}{\text{CSA of 2nd spherical solid}} = \dfrac{4πR^2}{4πr^2} \\[1em] = \dfrac{R^2}{r^2} \\[1em] = \Big(\dfrac{R}{r}\Big)^2 \\[1em] = \Big(\dfrac{3}{2}\Big)^2 \\[1em] = \dfrac{9}{4} \\[1em] = 9 : 4.$

Hence, Option 4 is the correct option.