How can the average density of the earth can be determined?

Step 1: Given & Assumptions

Considering earth as a perfect sphere of radius $\text{'}R\text{'}$

G is a gravitational constant.

g is the gravitational acceleration.

Step 2: Volume of Earth Formula

The volume of Earth $=\frac{4}{3}{\mathrm{\pi R}}^3$ (Vol. of the sphere)

Step 3: Finding the Mass of the earth.

$$\begin{gathered} M=volume\times averagedensity \\ e=averagedensity \end{gathered}$$

$$\begin{gathered} =v\times e \\ M=\frac{4}{3}{\mathrm{\pi R}}^3\times \mathrm{e} \end{gathered}$$

Step 4: Applying Newton's law of gravity

$$\begin{gathered} g=\frac{GM}{R^2} \\ g=\frac{G}{R^2}\left(\frac{4}{3}{\mathrm{\pi R}}^3\times \mathrm{e}\right) \end{gathered}$$ $\left\{puttingM=\frac{4}{3}{\mathrm{\pi R}}^3\times \mathrm{e}\right\}$

$g=\frac{4}{3}\mathrm{\pi RGe}$

Or $e=\frac{3g}{4\mathrm{\pi RG}}$

Hence, the average density of the earth is $e=\frac{3g}{4\mathrm{\pi RG}}$.