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?
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?
Solution:
Let the diameter of earth be $d$. Therefore, the radius of earth will be $\frac{d}{2}$.
Diameter of moon will be $\frac{d}{4}$ and the radius of moon will be $\frac{d}{8}$.
Volume of moon $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{8}\right)^{3}=\frac{1}{512} \times \frac{4}{3} \pi d^{3}$
Volume of earth $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{2}\right)^{3}=\frac{1}{8} \times \frac{4}{3} \pi d^{3}$
$\frac{\text { Volume of moon }}{\text { Volume of earth }}=\frac{\frac{1}{512} \times \frac{4}{3} \pi d^{3}}{\frac{1}{8} \times \frac{4}{3} \pi d^{3}}$
$=\frac{1}{64}$
$\Rightarrow$ Volume of moon $=\frac{1}{64}$ Volume of earth
Therefore, the volume of moon is $\frac{1}{64}$ of the volume of earth.
Let the diameter of earth be $d$. Therefore, the radius of earth will be $\frac{d}{2}$.
Diameter of moon will be $\frac{d}{4}$ and the radius of moon will be $\frac{d}{8}$.
Volume of moon $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{8}\right)^{3}=\frac{1}{512} \times \frac{4}{3} \pi d^{3}$
Volume of earth $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{2}\right)^{3}=\frac{1}{8} \times \frac{4}{3} \pi d^{3}$
$\frac{\text { Volume of moon }}{\text { Volume of earth }}=\frac{\frac{1}{512} \times \frac{4}{3} \pi d^{3}}{\frac{1}{8} \times \frac{4}{3} \pi d^{3}}$
$=\frac{1}{64}$
$\Rightarrow$ Volume of moon $=\frac{1}{64}$ Volume of earth
Therefore, the volume of moon is $\frac{1}{64}$ of the volume of earth.