Chapter 1, Problem 51GP

To determine

The volume of the moon.

Also, the number of moons needed to create a volume equal to the volume of Earth.

Answer to Problem 51GP

Volume of the moon is $2.2\times {10}^{19}{\text{m}}^3$ and about 49 moons are required to create a volume equal to that of the Earth.

Explanation of Solution

Given:

The diameter of the moon = 3480 km.

Formula used:

The volume of sphere is

  $V=\frac{4}{3}\pi r^3$

Where, r is the radius of the sphere.

Calculation : 

The moon is considered to be a sphere.

Diameter of the moon = 3480 km.

Radius of the moon is,

  $\begin{array}{c}r=\frac{3480}{2}\text{km}\\ \text{= }1740\text{ km }\\ =1,740,000\text{ m}\end{array}$

Therefore, the volume of the moon is

  $\begin{array}{c}V=\frac{4}{3}\pi r^3\\ =\frac{4}{3}\times \pi \times {1740000}^3\\ =2.2\times {10}^{19}{\text{m}}^3\end{array}$

The radius of the Earth is 6371 km.

Therefore, the volume of Earth is,

  $\begin{array}{c}V=\frac{4}{3}\pi r^3\\ =\frac{4}{3}\times \pi \times {6371000}^3{\text{m}}^3\end{array}$

To calculate the number moons required, divide the volume of the Earth by the volume of the moon.

So,

  $\begin{array}{c}N=\frac{\frac{4}{3}\times \pi \times {6371000}^3}{\frac{4}{3}\times \pi \times {1740000}^3}\\ =49.08\end{array}$

Conclusion:

Therefore, volume of Moon is $V=2.2\times {10}^{19}{\text{m}}^3$

Also, 49 moons would be required to create a volume equal to that of the Earth.