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

Question:

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

Solution:

Suppose that the radii of the spheres are r and R.

We have:

$\frac{4 \pi r^{2}}{4 \pi \mathrm{R}^{2}}=\frac{1}{4}$

$\Rightarrow \frac{r}{R}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

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

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

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