Answer
The center of mass of the earth-plus-moon system is 4670 km from the center of the earth.
Work Step by Step
Let the center of the earth be at the point x = 0. Let the center of the moon be at x = 384,000 km.
We can use $M_e = 5.97\times 10^{24}~kg$ as the mass of the earth. We can use $M_m = 7.35\times 10^{22}~kg$ as the mass of the moon.
We can find the center of mass $x_{cm}$ of the earth-plus-moon system.
$x_{cm} = \frac{(M_e)(0)+(M_m)(3.84\times 10^5~km)}{M_e+M_m}$
$x_{cm} = \frac{(5.97\times 10^{24}~kg)(0)+(7.35\times 10^{22}~kg)(3.84\times 10^5~km)}{5.97\times 10^{24}~kg+7.35\times 10^{22}~kg}$
$x_{cm} = 4670~km$
The center of mass of the earth-plus-moon system is 4670 km from the center of the earth.
