Answer
The aging rate of an astronaut is ${{R}_{a}}=0.44{{R}_{f}}$ and the astronaut is gone for 44weeks.
Work Step by Step
Substitute $v=0.9c$in Einstein’s special relativity equation and find ${{R}_{a}}$
$\begin{align}
& {{R}_{a}}={{R}_{f}}\sqrt{1-{{\left( \frac{0.9c}{c} \right)}^{2}}} \\
& ={{R}_{f}}\sqrt{1-0.81} \\
& ={{R}_{f}}\sqrt{0.19} \\
& =0.44{{R}_{f}}
\end{align}$
Now, use calculator to find $\sqrt{0.19}$
$\sqrt{0.19}=0.43588$
Now, substitute $100$ for ${{R}_{f}}$ and find ${{R}_{a}}$
$\begin{align}
& {{R}_{a}}=100\times \left( 0.43588 \right) \\
& =43.588\approx 44
\end{align}$
