First, the "connection" between the radius r (in meters) in a satellites trajectory and orbital period time is given by the following equation:
$\frac{r^3}{T^2}= \frac{k}{4\pi^2} $
$K$ is a constant equal to $3.98 \cdot 10^{14}$.
I solved for $r$ and $T$ because that will be needed.
$r=\sqrt[3]{\frac{T^2k}{4\pi^2}}$ and $T = \sqrt{\frac{r^3\pi^24}{k}} $
The question: Satellite $X$ runs in a trajectory twice the radius of Satellite $Y$. Which satellite has the longest turnaround, and how much longer is it.
My thoughts: Now I know that this should be solved algebraically, instead of just testing with numbers and predicting that way. Any suggestions on how to proceed from here? Would the following be a good "strategy"?:
$\sqrt{\frac{(2r)^3\pi^24}{k}} - \sqrt{\frac{r^3\pi^24}{k}} .$
My primary motivation for posting is debating and discussing math because I'm re-doing old exams on my own. It's nice to get some input.