Question

The diameter of the moon is approximately one-fourth of the diameter of the earth. Compare their volume and curved surface area.

Answer 1

Let the diameter of earth be $d$ and that of the moon will be $\frac{d}{4}$
Radius of the earth $r_e=\frac{d}{2}$
Radius of the moon $r_m=\frac{d}{8}$

We know that surface area of sphere $=4\pi r^2$
Ratio of their surface area $=\frac{\text{Surface area of Moon}}{\text{Surface area of Earth}}=\frac{4\pi {r_m}^2}{4\pi {r_e}^2}$
$=\frac{4\pi {\left(\frac{d}{8}\right)}^2}{4\pi {\left(\frac{d}{2}\right)}^2}$
$=\frac{\frac{1}{64}}{\frac{1}{4}}$
$=\frac{4}{64}$
$=\frac{1}{16}$

Thus, the ratio of their surface areas is $1:16$.

Similarly, volume of a sphere is $\frac{4}{3}\pi r^3$

$\therefore \frac{\text{Volume of Moon}}{\text{Volume of Earth}}=\frac{\frac{4}{3}\pi r_m^3}{\frac{4}{3}\pi r_e^3}$

$=\frac{r_m^3}{r_e^3}$

$=\frac{(\frac{d}{8}{)}^3}{(\frac{d}{2}{)}^3}$

$=(\frac{d}{8}{)}^3\times (\frac{2}{d}{)}^3$

$=\frac{8}{512}$

$=\frac{1}{64}$

Thus, the ratio of their volumes is $1:64$.