Question

The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Answer 1

Step 1: Find the volume of earth and moon.

Let the diameter of earth $=dkm$, so the radius $\left(r_1\right)=\frac{d}{2}km$

Therefore diameter of moon $=\frac{d}{4}km$ and radius of moon $\left(r_2\right)=\frac{d}{8}km$

Volume of the earth $=\frac{4}{3}{{\mathrm{\pi r}}_1}^3$ [As the volume of sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$]

$$\begin{gathered} =\frac{4}{3}\mathrm{\pi x}{\left(\frac{\mathrm{d}}{2}\right)}^2 \\ =\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{d}}^3}{8}\right)..............................\left(\mathrm{i}\right) \end{gathered}$$

Similarly, Volume of the moon $=\frac{4}{3}{{\mathrm{\pi r}}_2}^3$

$$\begin{gathered} =\frac{4}{3}\mathrm{\pi x}{\left(\frac{\mathrm{d}}{8}\right)}^2 \\ =\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{d}}^3}{512}\right)..............................\left(\mathrm{i}\right) \end{gathered}$$

Step 2: Finding the volume of moon in terms of the volume of the earth.

$$\begin{gathered} \frac{Volumeofmoon}{volumeofearth}=\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }\left({\displaystyle \frac{{\mathrm{d}}^3}{512}}\right)}{\frac{4}{3}\mathrm{\pi }\left(\frac{{\mathrm{d}}^3}{8}\right)} \\ \Rightarrow \frac{Volumeofmoon}{volumeofearth}=\frac{{\displaystyle 1}}{64} \\ \Rightarrow volumeofmoon=\frac{{\displaystyle 1}}{64}ofvolumeofearth \end{gathered}$$

Hence , the volume of the moon is $\frac{1}{{64}^{th}}$ of the volume of the earth.