Answer
$4.67\times 10^{6}\mathrm{m}$ from the center of Earth, which is
$1.70\times 10^{6}\mathrm{m}$ below the surface of the Earth
Work Step by Step
Use equation 9-14 $\quad X_{\mathrm{c}\mathrm{m}}=\displaystyle \frac{\sum mx}{M}$
Let the origin be at the center of the Earth.
Earth: $m_{1}=5.98\times 10^{24}kg, \qquad x_{1}=0 m$
Moon:$ m_{2}=7.35\times 10^{22}kg,\qquad x_{2}=3.85\times 10^{8}m$
$X_{\mathrm{c}\mathrm{m}}=\displaystyle \frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$
$=\displaystyle \frac{0+(5.98\times 10^{24}kg)(3.85\times 10^{8}m)}{5.98\times 10^{24}kg+7.35\times 10^{22}kg}$
$=4.67\times 10^{6}\mathrm{m}$
$R_{\mathrm{E}}=6.37\times 10^{6}\mathrm{m}$ so $X_{\mathrm{c}\mathrm{m}}$ is below the surface of the Earth,
$6.37\times 10^{6}\mathrm{m}-4.67\times 10^{6}\mathrm{m}=1.70\times 10^{6}\mathrm{m}$
below the surface of the Earth.
