Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radii}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{spheres}\mathrm{be}r\mathrm{and}R. \\ \mathrm{As}, \\ \frac{\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{sphere}}{\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere}}=\frac{4}{25} \\ \Rightarrow \frac{4\mathrm{\pi }{r}^2}{4\mathrm{\pi }{R}^2}=\frac{4}{25} \\ \Rightarrow {\left(\frac{r}{R}\right)}^2=\frac{4}{25} \\ \Rightarrow \frac{r}{R}=\sqrt{\frac{4}{25}} \\ \Rightarrow \frac{r}{R}=\frac{2}{5}.....\left(\mathrm{i}\right) \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{volumes}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{spheres}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere}} \\ =\frac{\left({\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3\right)}{\left({\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3\right)} \\ ={\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^3 \\ ={\left(\frac{2}{5}\right)}^3\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ =\frac{8}{125} \\ =8:125 \end{gathered}$$

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