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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 :

Let the diameter of earth be d.
Therefore, the diameter of moon will be \(\frac{d}{4}\).
Thus, Radius of earth = \(\frac{d}{2}\)
\(\therefore \) Radius of moon
= \(\frac{1}{2}\) × \(\frac{d}{4}\)
= \(\frac{d}{8}\)

Now, Volume of moon
= \(\frac{4}{3}{\pi}{r}^3 \)
= \(\frac{4}{3}{\pi} × {\frac{d}{8}}^3 \)
=\( \frac{1}{512} × \frac{4}{3}{\pi} × {d}^3\)

Similarly,
Volume of earth
= \(\frac{4}{3}{\pi}{r}^3 \)
=\( \frac{4}{3}{\pi} × {\frac{d}{2}}^3 \)
= \(\frac{1}{8} × \frac{4}{3}{\pi} × {d}^3\)

\(\therefore \) \(\frac{Volume of moon}{Volume of earth} \)
=\( \frac{\frac{1}{512} × \frac{4}{3}{\pi} × {d}^3}{\frac{1}{8} × \frac{4}{3}{\pi} × {d}^3} \)
= \(\frac{1}{64}\)

Therefore, the volume of moon = \(\frac{1}{64}\) is of the volume of earth.

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