Answer
The true weight of the satellite on the planet is $24511.5N$.
Work Step by Step
To find the weight of the satellite on the planet's surface, we need to find the planet's gravitational acceleration $g$ on its surface, which is calculated by $$g=\frac{GM}{r^2} (1)$$
The planet's radius $r=4.15\times10^6m$. $GM$ is not known yet, so we need to find it.
The formula for a satellite's orbital period over a planet is $$T=\frac{2\pi r^{3/2}}{\sqrt{GM}}$$ $$GM=\Big(\frac{2\pi r^{3/2}}{T}\Big)^2=\frac{4\pi^2r^3}{T^2}$$
We know $T=2hr=7.2\times10^3s$. The distance from the satellite to the planet's core is $r=4.1\times10^5+4.15\times10^6=4.56\times10^6m$. Therefore,
$$GM=7.22\times10^{13}Nm^2/kg$$
Plug $GM$ back to (1) to find $g$, we have $$g=4.19m/s^2$$
Now we can calculate the weight of the satellite: $mg=5850\times4.19=24511.5N$
