Question

Volumes of two spheres are in the ratio 64 ; 27 . The ratio of their surface areas is
(a) 3 : 4 (b) 4 : 3 (c) 9 : 16 (d) 16 : 9

Answer 1

Volume of the spheres are in the ratio 64 : 27.
$$\begin{gathered} \Rightarrow V_1:V_2=64:27 \\ \Rightarrow \frac{4}{3}{\mathrm{\pi r}}_1^3:\frac{4}{3}{\mathrm{\pi r}}_2^3=64:27 \\ \Rightarrow {\mathrm{r}}_1^3:{\mathrm{r}}_2^3=64:27 \\ \Rightarrow {\left(\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\right)}^3=\frac{64}{27} \\ \Rightarrow {\left(\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\right)}^3={\left(\frac{4}{3}\right)}^3 \end{gathered}$$
Thus, ratio of the radii = 4 : 3
Ratio of the surface area of the spheres will be
$$\begin{gathered} \frac{4{\mathrm{\pi r}}_1^2}{4{\mathrm{\pi r}}_2^2} \\ ={\left(\frac{{\displaystyle r_1}}{{\displaystyle r_2}}\right)}^2={\left(\frac{4}{3}\right)}^2=\frac{16}{9} \end{gathered}$$
So, the ratio is 16 : 9.
Hence the correct answer is option (d).