Question

The diameter of the moon is approximately one fourth the diameter of the earth. Find the ratio of their surface areas.

Answer 1

Find the ratio of the surface areas of moon to earth.

Let $d_m$ be the diameter of the moon and $d_e$ be the radius of the earth.

Assuming both of them a spherical body.

It is given that $d_m=\frac{d_e}{4}$

The surface area of the sphere $=4{\mathrm{\pi r}}^2$

$\therefore \frac{Surfaceareaofthemoon}{Surfaceareaoftheearth}=\frac{4\mathrm{\pi }{\left({\displaystyle \frac{{\mathrm{d}}_{\mathrm{m}}}{2}}\right)}^2}{4\mathrm{\pi }{\left(\frac{{\mathrm{d}}_e}{2}\right)}^2}$

$\Rightarrow \frac{Surfaceareaofthemoon}{Surfaceareaoftheearth}=\frac{{\left({\displaystyle \frac{{\mathrm{d}}_e}{2\times 4}}\right)}^2}{{\left(\frac{{\mathrm{d}}_e}{2}\right)}^2}$

$\Rightarrow \frac{Surfaceareaofthemoon}{Surfaceareaoftheearth}=\frac{{\left({\displaystyle \frac{{\mathrm{d}}_e}{8}}\right)}^2}{{\left(\frac{{\mathrm{d}}_e}{2}\right)}^2}$

$\Rightarrow \frac{Surfaceareaofthemoon}{Surfaceareaoftheearth}=\frac{1}{16}$

Hence, the required ratio is $\frac{1}{16}$.