Answer
$\Delta t = \frac{r}{c-r~\alpha}$
Work Step by Step
The total distance traveled by the light wave is $~~r+r\alpha~\Delta t$
The light wave travels at a speed of $c$
We can find an expression for the required time:
$\Delta t = \frac{r+r~\alpha~\Delta t}{c}$
$c~\Delta t-r~\alpha~\Delta t = r$
$\Delta t = \frac{r}{c-r~\alpha}$
