Question

Choose the correct answer of the following question:

The surface areas of two spheres are in the ratio 16 : 9. The ratio of their volumes is

(a) 64 : 27 (b) 16 : 9 (c) 4 : 3 (d) 16³ : 9³ [CBSE 2013C]

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{spheres}\mathrm{be}r\mathrm{and}R. \\ \mathrm{As}, \\ \frac{\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{sphere}}{\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere}}=\frac{16}{9} \\ \Rightarrow \frac{4\mathrm{\pi }{R}^2}{4\mathrm{\pi }{r}^2}=\frac{16}{9} \\ \Rightarrow {\left(\frac{R}{r}\right)}^2=\frac{16}{9} \\ \Rightarrow \frac{R}{r}=\sqrt{\frac{16}{9}} \\ \Rightarrow \frac{R}{r}=\frac{4}{3}.....\left(\mathrm{i}\right) \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{their}\mathrm{volumes}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere}} \\ =\frac{\left({\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3\right)}{\left({\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^3\right)} \\ ={\left(\frac{R}{r}\right)}^3 \\ ={\left(\frac{4}{3}\right)}^3\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ =\frac{64}{27} \\ =64:27 \end{gathered}$$

Hence, the correct answer is option (a).