Question

A solid sphere and a solid hemisphere have the same total surface area. Find the ratio of their volumes.

Answer 1

Let $R$ and $r$ be the radius of the solid sphere and the solid hemisphere.

Now the surface area of the solid sphere and the solid hemisphere are $4\pi R^2$ and $3{\pi }^2$.

According to the problem,

$4\pi R^2=3{\pi }^2$

or, $4R^2=3r^2$

or, $R=\frac{\sqrt{3}r}{2}$......(1).

Now ratio of their volumes is $\frac{\frac{4}{3}\pi R^3}{\frac{2}{3}\pi r^3}=\frac{2R^3}{r^3}=\frac{3\sqrt{3}{r}^3}{4r^3}=\frac{3\sqrt{3}}{4}$ [Using (1)].