Question

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

Answer 1

Let the side of the cube$=a$

So, the surface area of the cube$=6a^2$

Let the radius of the sphere$=r$

So, the surface area of the sphere $=4\pi r^2$

Therefore,

$6a^2=4\pi r^2$ $=x$(let)

Therefore,

$a=\sqrt{\frac{x}{6}}$

And $r=\sqrt{\frac{x}{4\pi }}$

Therefore, ratio of their volumes,

$=\frac{a^3}{\frac{4}{3}\pi r^3}$

$=\frac{{\left(\sqrt{\frac{x}{6}}\right)}^3}{\frac{4}{3}\pi {\left(\sqrt{\frac{x}{4\pi }}\right)}^3}$

$=\frac{\frac{1}{6}\sqrt{\frac{1}{6}}}{\frac{4}{3}\pi \frac{1}{4\pi }\sqrt{\frac{1}{4\pi }}}$

$=\frac{\frac{1}{2}\sqrt{4\pi }}{\sqrt{6}}$

$=\sqrt{\frac{4\pi }{24}}$

$=\frac{\sqrt{\pi }}{\sqrt{6}}$

So, the ratio is $\sqrt{\pi }:\sqrt{6}$.