Answer
$5\times10^{-10}\;s$
Work Step by Step
The distance between the Earth and the Moon is: $ d= 3.82\times10^8\;m$
Therefore, the predicted value of the time for the laser to travel from an Earth-bound observatory to a reflector on the Moon is
$t=\frac{2d}{c}$
Taking $\ln$ in both side, we obtain
$\ln t=\ln(\frac{2d}{c})$
or, $\ln t=\ln(2d)-\ln c$
or, $\ln t=\ln 2+\ln(d)-\ln c$
or, $\frac{\delta t}{t}=\frac{\delta d}{d}$
or, $\delta t=\frac{\delta dt}{d}$
Here,
$2\delta d=15\;cm$
or, $\delta d=0.15\;m$
or, $\delta d=0.075\;m$
Putting known values, we batin
or, $\delta t=\frac{0.075\times2.55}{3.82\times10^8}\;s$
or, $t=5\times10^{-10}\;s$
Therefore, the uncertainty is $5\times10^{-10}\;s$
