Given that the period of the Moon's orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84 108 m, use T2 = 4 π2 GME a3 to calculate the mass of the Earth.

M = 6.014x10^24 kg

Explanation:

First of all we need to gather all dа ta:

Period (T) is 27.32 days

Constant distant (a) is 3.84x10^8 m

The expression given is T² = (4π²/GM) * a³

We need to know the value of the constant G which is 6.674x10^-11 Nm² / kg²

Finally M is the mass of the earth.

From that expression, we should solve for M:

M = (4π²/GT²) * a³

In this case, we just trade the T for the M, because it was the only change we needed to do. Now, before we can do the calculations, let's convert the days to second:

T = 27.32 days * 24 h/day * 3600 s/h = 2,360,448 s

Now, let's solve for M replacing all the data in the formula:

M = (4π² / 6.674x10^-11 * 2,360,448²) * (3.84x10^8) ³

M = (39.4784 / 371.6891) * 5.662x10^25

M = 6.014x10^24 kg