Answer
The mass of the planet is $1.01\times 10^{26}~kg$
Work Step by Step
We can find the distance $d$ that the moon travels when it completes one orbit.
$d = v~t$
$d = (7500~m/s)(28~hr)(3600~s/hr)$
$d = 7.56\times 10^8~m$
We can find the orbital radius $R$.
$2\pi R = d$
$R =\frac{d}{2\pi}$
$R = \frac{7.56\times 10^8~m}{2\pi}$
$R = 1.20\times 10^8~m$
We can use the equation for orbital speed to find the mass of the planet.
$v = \sqrt{\frac{G~M}{R}}$
$v^2 = \frac{G~M}{R}$
$M = \frac{v^2~R}{G}$
$M = \frac{(7500~m/s)^2(1.20\times 10^8~m)}{6.67\times 10^{-11}~m^3/kg~s^2}$
$M = 1.01\times 10^{26}~kg$
The mass of the planet is $1.01\times 10^{26}~kg$
