Question:
If the ratio of the volumes of two spheres is 1 : 8, then the ratio of their surface area is
(a) 1 : 2
(b) 1 : 4
(c) 1 : 8
(d) 1 : 16
Solution:
(b) 1 : 4
Suppose that r and R are the radii of the spheres.
Then we have:
$\frac{\frac{4}{3} \pi r^{3}}{\frac{4}{3} \pi R^{3}}=\frac{1}{8}$
$\Rightarrow\left(\frac{r}{R}\right)^{3}=\frac{1}{8}$
$\Rightarrow \frac{r}{R}=\frac{1}{2}$
$\therefore$ Ratio of surface area of spheres $=\frac{4 \pi r^{2}}{4 \pi R^{2}}=\left(\frac{r}{R}\right)^{2}=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$