Step 1: Volume Formula of a Sphere
The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.
Step 2: Relating the Volumes of the Moon and Earth
Let the radius of the Earth be \( r_{\text{earth}} \) and the radius of the Moon be \( r_{\text{moon}} \). Since the diameter of the Moon is one-fourth of the diameter of the Earth, we have: \[ r_{\text{moon}} = \frac{1}{4} r_{\text{earth}} \]
Step 3: Volume of the Moon and the Earth
The volume of the Earth is: \[ V_{\text{earth}} = \frac{4}{3} \pi r_{\text{earth}}^3 \] The volume of the Moon is: \[ V_{\text{moon}} = \frac{4}{3} \pi r_{\text{moon}}^3 = \frac{4}{3} \pi \left(\frac{1}{4} r_{\text{earth}}\right)^3 = \frac{4}{3} \pi \frac{1}{64} r_{\text{earth}}^3 \]
Step 4: Fraction of the Volume
To find the fraction of the volume of the Earth that is the volume of the Moon, we divide the volume of the Moon by the volume of the Earth: \[ \frac{V_{\text{moon}}}{V_{\text{earth}}} = \frac{\frac{4}{3} \pi \frac{1}{64} r_{\text{earth}}^3}{\frac{4}{3} \pi r_{\text{earth}}^3} \] Simplifying: \[ \frac{V_{\text{moon}}}{V_{\text{earth}}} = \frac{1}{64} \]
The volume of the Moon is \( \frac{1}{64} \) of the volume of the Earth.