Answer
3.76 meters per century
Work Step by Step
We follow the book's recommendation to differentiate equation 8.4. We obtain:
$T^2 = \frac{4\pi^2r^3}{GM}$
$2T (\frac{DT}{dt})=\frac{12\pi^2r^2}{GM}\times \frac{dr}{dt}$
Recall, the orbital period of the moon is 27.3 days, which equals $2.359\times10^6s$. The distance between the center of the earth and the moon is $384,400\times10^3 m$. Thus, we find:
$\frac{dr}{dt}=\fbox{3.76 meters per century}$
