Volumes of two spheres are in the ratio $64:27$. Find the ratio of their surface area.

Given: Volumes of two spheres are in the ratio $64:27$.

To do: To find the ratio of their surface area.

To find the ratio of the surface areas, first we have to find the surface areas with their volumes.

Radius of big sphere $=R$

Radius of small sphere $=r$  

Volume of bigger sphere$=\frac{4}{3}\pi R^3$

Volume of smaller sphere$=\frac{4}{3}\pi r^3$

Given, Volume of bigger sphere$:$Volume of smaller sphere $=64:27$

$\Rightarrow \frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3}=\frac{64}{27}$

$\Rightarrow \frac{R^3}{r^3}=\frac{64}{27}$

$\Rightarrow \frac{R}{r}=\sqrt[3]{\frac{64}{27}}$

$\Rightarrow \frac{R}{r}=\frac{4}{3}$

Surface area of bigger sphere$=4\pi R^2$

Surface area of smaller sphere$=4\pi r^2$ 

Hence, Surface area of bigger sphere: Surface area of smaller sphere$=\frac{4\pi R^2}{4\pi r^2}$

$=( \frac{R}{r})^2$

$=( \frac{4}{3})^2$

Thus, the ratio of their surface areas $=16:9$