Question

If the ratio of the volumes of two spheres is $8:27$, then the ratio of their surface areas is

Answer 1

Volume of first sphere $=\frac{4}{3}\pi r_1^3$
and volume of second sphere $=\frac{4}{3}\pi r_2^3$
But ratio in their volumes $=8:27$
$\therefore \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}=\frac{8}{27}$
$\Rightarrow \frac{r_1^3}{r_2^3}=\frac{8}{27}={\left(\frac{2}{3}\right)}^3$
$\Rightarrow {\left(\frac{r_1}{r_2}\right)}^3={\left(\frac{2}{3}\right)}^3$
$\Rightarrow \frac{r_1}{r_2}=\frac{2}{3}$
Now, ratio in their surface areas $=\frac{4\pi r_1^3}{4\pi r_2^3}$
$=\frac{r_1^2}{r_2^2}={\left(\frac{r_1}{r_2}\right)}^2={\left(\frac{2}{3}\right)}^2=\frac{4}{9}$
$\therefore$ Ratio is $4:9$
Hence, the ratio of their volumes is $4:9$