Question

The volumes of the two spheres are in the ratio $64:27.$ Find the ratio of their surface areas.

Answer 1

Let the radius of two spheres be $r_1$ and $r_2$.
Given, the ratio of the volume of two spheres $=64:27$

$\frac{V_1}{V_2}=\frac{64}{27}$

$\Rightarrow \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}=\frac{64}{27}$ $[\because$ volume of sphere $=\frac{4}{3}\pi r^3]$

$\Rightarrow {\left(\frac{r_1}{r_2}\right)}^3={\left(\frac{4}{3}\right)}^3$

$\Rightarrow \frac{r_1}{r_2}=\frac{4}{3}$

Let the surface areas of the two spheres be $S_1$ and $S_2$.

$\therefore \frac{S_1}{S_2}=\frac{4\pi r_1^2}{4\pi r_2^2}={\left(\frac{r_1}{r_2}\right)}^2$ [$\because$ Surface area of sphere $=4\pi r^2$]

$\Rightarrow S_1:S_2={\left(\frac{4}{3}\right)}^2=\frac{16}{9}$

$\Rightarrow S_1:S_2=16:9$

Hence, the ratio of their surface areas is $16:9.$