Answer
The trip takes a time of 8.9 hours according to the Rolex.
Work Step by Step
Let $\Delta t_0$ be the time measured on the Rolex.
Let $T$ be the time measured at mission control.
We can find the time measured on the Rolex:
$T = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}}$
$\Delta t_0 = T~\sqrt{1-\frac{v^2}{c^2}}$
$\Delta t_0 = T~\sqrt{1-\frac{(2.0\times 10^8~m/s)^2}{(3.0\times 10^8~m/s)^2}}$
$\Delta t_0 = (12.0~h)~\sqrt{\frac{5}{9}}$
$\Delta t_0 = 8.9~hours$
The trip takes a time of 8.9 hours according to the Rolex.
