Answer
The aging rate of astronaut is $0.6{{R}_{f}}$ and the astronaut is gone for $60$ weeks.
Work Step by Step
Consider the provided formula,
${{R}_{a}}={{R}_{f}}\sqrt{1-{{\left( \frac{v}{c} \right)}^{2}}}$
Substitute $v=0.8c$ in equation (I) and find ${{R}_{a}}$, that is.,
$\begin{align}
& {{R}_{a}}={{R}_{f}}\sqrt{1-{{\left( \frac{0.8c}{c} \right)}^{2}}} \\
& ={{R}_{f}}\sqrt{1-0.64} \\
& ={{R}_{f}}\sqrt{0.36} \\
& =0.6{{R}_{f}}
\end{align}$
Therefore, the aging rate is${{R}_{a}}=0.6{{R}_{f}}$
Now, to find the number of weeks for which astronaut is gone.
Substitute $100$ for ${{R}_{f}}$ in equation (I) and find ${{R}_{a}}$, that is.,
$\begin{align}
& {{R}_{a}}=100\times \left( 0.6 \right) \\
& =60
\end{align}$
