Question

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

Answer 1

(b) 16 : 9
Let the radii of the spheres be R and r.
Then, ratio of their volumes
$=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}$

Therefore,

$$\begin{gathered} \frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}=\frac{64}{27} \\ \Rightarrow \frac{R^3}{r^3}=\frac{64}{27} \end{gathered}$$

$$\begin{gathered} \Rightarrow {\left(\frac{R}{r}\right)}^3={\left(\frac{4}{3}\right)}^3 \\ \Rightarrow \frac{R}{r}=\frac{4}{3} \end{gathered}$$
Hence, the ratio of their surface areas$=\frac{4\mathrm{\pi }{R}^2}{4{\mathrm{\pi r}}^2}$

$$\begin{gathered} =\frac{R^2}{r^2} \\ ={\left(\frac{R}{r}\right)}^2 \\ ={\left(\frac{4}{3}\right)}^2 \\ =\frac{16}{9} \\ =16:9 \end{gathered}$$