Question

The diameter of the moon is approximately one - fourth the diameter of earth. What fraction of the volume of earth is the volume of the moon?
(a). $\dfrac{1}{4}$
(b). $\dfrac{1}{16}$
(c). $\dfrac{1}{64}$
(d). $\dfrac{1}{256}$

Answer 1

Hint: First from the given information we are going to find the relation between the radius of earth and moon and then we will use the formula for calculating the volume of the sphere and then we can find the ratio of volume.

Complete step-by-step solution -

Let’s start by writing the relation between the radius of earth and moon.
${{d}_{1}}$ = diameter of earth
${{d}_{2}}$ = diameter of moon
${{r}_{1}}$ = radius of earth
${{r}_{2}}$ = radius of moon
Now as per the question,
${{d}_{2}}$= $\dfrac{{{d}_{1}}}{4}$
And we also know that,
${{d}_{1}}$= 2${{r}_{1}}$ and ${{d}_{2}}$= 2${{r}_{2}}$
Using this we get,
\[2{{r}_{2}}=\dfrac{2{{r}_{1}}}{4}\]
\[{{r}_{2}}=\dfrac{{{r}_{1}}}{4}\]
Now the volume of sphere having radius r is: $\dfrac{4\pi {{r}^{3}}}{3}$
Now we will use this to find the ratio of volumes of earth and moon,
Let ${{v}_{1}}$ = volume of earth
Let ${{v}_{2}}$ = volume of moon
Now $\dfrac{{{v}_{2}}}{{{v}_{1}}}$ is :
$=\dfrac{\dfrac{4\pi {{r}_{2}}^{3}}{3}}{\dfrac{4\pi {{r}_{1}}^{3}}{3}}$
Cancelling the common terms we get,
Now using \[{{r}_{2}}=\dfrac{{{r}_{1}}}{4}\] we get,
$\begin{align}
  & ={{\left( \dfrac{{{r}_{2}}}{{{r}_{1}}} \right)}^{3}} \\
 & ={{\left( \dfrac{{{r}_{1}}}{4{{r}_{1}}} \right)}^{3}} \\
 & =\dfrac{1}{{{4}^{3}}} \\
 & =\dfrac{1}{64} \\
\end{align}$
Hence the ratio of volume of moon by volume earth or $\dfrac{{{v}_{2}}}{{{v}_{1}}}$ is $\dfrac{1}{64}$ .
So, option (c) is correct.

Note: We can also solve this question by using the formula for volume of sphere in terms of diameter and then by using the given relation we can find the ratio of volume of moon by the volume of earth directly with using the relation of radius and diameter.