The surface areas of two spheres are in the ratio of 4 : 25.
Question:

The surface areas of two spheres are in the ratio of 4 : 25. Find the ratio of their volumes.

Solution:

Let the radii of the two spheres be $r$ and $R$.

As,

$\frac{\text { Surface area of the first sphere }}{\text { Surface area of the second sphere }}=\frac{4}{25}$

$\Rightarrow \frac{4 \pi r^{2}}{4 \pi R^{2}}=\frac{4}{25}$

$\Rightarrow\left(\frac{r}{R}\right)^{2}=\frac{4}{25}$

$\Rightarrow \frac{r}{R}=\sqrt{\frac{4}{25}}$

$\Rightarrow \frac{r}{R}=\frac{2}{5} \quad \ldots \ldots(\mathrm{i})$

Now,

The ratio of the volumes of the two spheres $=\frac{\text { Volume of the first sphere }}{\text { Volume of the second sphere }}$

$=\frac{\left(\frac{4}{3} \pi r^{3}\right)}{\left(\frac{4}{3} \pi R^{3}\right)}$

$=\left(\frac{\mathrm{r}}{\mathrm{R}}\right)^{3}$

$=\left(\frac{2}{5}\right)^{3} \quad[\operatorname{Using}(\mathrm{i})]$

$=\frac{8}{125}$

$=8: 125$

So, the ratio of the volumes of the given spheres is 8 : 125.

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