#### Answer

(a) The center of mass is a distance of $4.66\times 10^6~m$ from the center of the Earth.
(b) The center of mass of the Earth-Moon system moves smoothly in an ellipse around the Sun. The Moon revolves around the center of mass of the Earth-Moon system as the Earth-Moon system goes smoothly around the Sun. The Earth wobbles around the center of mass of the Earth-Moon system as the Earth-Moon system goes smoothly around the Sun.

#### Work Step by Step

(a) Let the point $x=0$ be the center of the Earth. Let $m_E$ be the mass of the Earth. Let $m_M$ be the mass of the moon.
$x_{CM} = \frac{m_E~x_E+m_M~x_M}{m_E+m_M}$
$x_{CM} = \frac{0+(7.35\times 10^{22}~kg)(3.84\times 10^8~m)}{5.98\times 10^{24}~kg+7.35\times 10^{22}~kg}$
$x_{CM} = 4.66\times 10^6~m$
The center of mass is a distance of $4.66\times 10^6~m$ from the center of the Earth. (Note that this is still inside the Earth since the Earth's radius is $6.38\times 10^6~m$)
(b) The center of mass of the Earth-Moon system moves smoothly in an ellipse around the Sun. The Moon revolves around the center of mass of the Earth-Moon system as the Earth-Moon system goes smoothly around the Sun. The Earth wobbles around the center of mass of the Earth-Moon system as the Earth-Moon system goes smoothly around the Sun.