Question

The surface areas of two spheres are in the ratio 1 : 4. Find the ratio of their volumes.

Answer 1

Suppose that the radii of the spheres are r and R.
We have:
$$\begin{gathered} \frac{4\mathrm{\pi }{r}^2}{4{\mathrm{\pi R}}^2}=\frac{1}{4} \\ \Rightarrow \frac{r}{R}=\sqrt{\frac{1}{4}}=\frac{1}{2} \end{gathered}$$

Now, ratio of the volumes = $\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3}{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}={\left(\frac{r}{R}\right)}^3={\left(\frac{1}{2}\right)}^3=\frac{1}{8}$

∴ The ratio of the volumes of the spheres is 1 : 8.