Answer
See below.
Work Step by Step
I substitute in $v=0.9c$ in the given expression:
Thus $R_a=R_f\sqrt{1-(\frac{v}{c})^2}=R_f\sqrt{1-(\frac{0.9c}{c})^2}=R_f\sqrt{1-(0.9)^2}=R_f\sqrt{1-0.81}=R_f\sqrt{0.19}\approx0.44R_f$
Thus the aging rate relative to someone's aging rate on Earth is $0.44$
Thus $44 $ weeks for me are $\frac{44}{0.44}=100$ weeks for my friend.
