Question

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

Options

• 27 : 8

• 8 : 27

• 3 : 2

• 9 : 4

Answer 1

9 : 4

Explanation:

Let radius of two spherical solids be R and r.

Given,

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

\[\Rightarrow \frac{\text{Vol. of 1st spherical solid}}{\text{Vol. of 2nd spherical solid}} = \frac{27}{8}\]

\[\Rightarrow \frac{\frac{4}{3}\, \pi R^3}{\frac{4}{3}\, \pi r^3} = \frac{27}{8}\]

\[\Rightarrow \frac{R^3}{r^3} = \frac{27}{8}\]

\[\Rightarrow \left(\frac{R}{r}\right)^3 = \left(\frac{3}{2}\right)^3\]

\[\Rightarrow \frac{R}{r} = \frac{3}{2} \; \text{.........(1)}\]

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

\[\Rightarrow \frac{\text{CSA of 1st spherical solid}}{\text{CSA of 2nd spherical solid}} = \frac{4\pi R^2}{4\pi r^2}\]

\[{} = \frac{R^2}{r^2}\]

\[{} = \left(\frac{R}{r}\right)^2\]

\[{} = \left(\frac{3}{2}\right)^2\]

\[{} = \frac{9}{4}\]

= 9 : 4