A satellite is orbiting just above the surface of the earth with...

The formula to calculate the mass of the Earth considering it a sphere is given by

$M = \dfrac{4}{3}\pi{R}^{3}d . . . \left(1\right)$

The formula to calculate the time period of revolution of the satellite can be written as

$T = 2\pi\sqrt{\dfrac{{R}^{3}}{GM}} . . . \left(2\right)$

Substitute the expression for the mass from equation (1) into equation (2) and simplify to obtain the expression for the required quantity.

$\begin{cases}T & = & 2\pi\sqrt{\dfrac{{R}^{3}}{G\dfrac{4}{3}\pi{R}^{3}d}} \\ & = & 2\pi\sqrt{\dfrac{3}{4\piGd}} \\ & = & \sqrt{\dfrac{12{\pi}^{2}}{4\piGd}} \\ & = & \sqrt{\dfrac{3\pi}{Gd}} \\ & ⇒ & \dfrac{3\pi}{Gd} = {T}^{2}\end{cases}$