The surface areas of a sphere and a cube are equal.
Question:

The surface areas of a sphere and a cube are equal. Find the ratio of their volumes.

Solution:

Surface area of the sphere $=4 \pi r^{2}$

Surface area of the cube $=6 a^{2}$

Therefore,

$4 \pi r^{2}=6 a^{2}$

$\Rightarrow 2 \pi r^{2}=3 a^{2}$

$\Rightarrow r^{2}=\frac{3 a^{2}}{2 \pi}$

$\Rightarrow r=\sqrt{\frac{3}{2 \pi}} a$

Ratio of their volumes $=\frac{4 / 3 \pi r^{3}}{a^{3}}=\frac{4 \pi r^{3}}{3 a^{3}}$

$=\frac{4 \pi}{3 a^{3}} \times \frac{3 \sqrt{3} a^{3}}{2 \pi \sqrt{2 \pi}} \quad\left[\right.$ Since $\left.r=\sqrt{\frac{3}{2 \pi}} a\right]$

$=\frac{2 \sqrt{3}}{\sqrt{2 \pi}}$

$=\frac{2 \times \sqrt{3}}{\sqrt{2} \times \frac{\sqrt{22}}{\sqrt{7}}}$

$=\frac{2 \times \sqrt{3} \times \sqrt{7}}{\sqrt{2} \times \sqrt{2} \times \sqrt{11}}$

$=\frac{\sqrt{21}}{\sqrt{11}}$

Thus, the ratio of their volumes is $\sqrt{21}: \sqrt{11}$.