Question

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

• A. $1:2$
• B. $1:4$
• C. $1:8$
• D. $1:16$

Answer 1

The correct option is B

$1:4$

Explanation for the correct option:

Find the ratio of radius of the two spheres

Given, the ratio of the volumes of two spheres is $1:8$

The volume of a sphere of radius $r$ $=\frac{4}{3}{\mathrm{\pi r}}^3$

Let us consider $r_1$ and $r_2$ be the radii of the two spheres.

The ratio of the volume of the two spheres$=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^3}{\frac{4}{3}\mathrm{\pi }{\left({\mathrm{r}}_2\right)}^3}=\frac{1}{8}$

$\Rightarrow \frac{{\mathrm{r}}_1^3}{{\mathrm{r}}_2^3}=\frac{1}{8}$

$\Rightarrow \frac{{\mathrm{r}}_1^3}{{\mathrm{r}}_2^3}=\frac{1^3}{2^3}$ $\left(\because \begin{array}{c}{1}^3=1\\ 2^3=8\end{array}\right)$

$\Rightarrow \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{1}{2}$

Find the ratio of surface areas of the two spheres

The surface area of sphere of radius $r$ $=4{\mathrm{\pi r}}^2$

Now, ratio of surface area of the two spheres$=\frac{{\displaystyle 4}\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{4\mathrm{\pi }{\left({\mathrm{r}}_2\right)}^2}$

$=\frac{{\left({\mathrm{r}}_1\right)}^2}{{\left({\mathrm{r}}_2\right)}^2}$

$={\left(\frac{r_1}{r_2}\right)}^2$ $\left(\because \frac{a^m}{b^m}={\left(\frac{a}{b}\right)}^m\right)$

$={\left(\frac{1}{2}\right)}^2$

$=\frac{1^2}{2^2}$

$=\frac{1}{4}$

Therefore, the required ratio of surface areas of two spheres is $1:4$.

Hence, option B is correct.