Answer
The distance between the galaxy and the Earth when the light was emitted is $~~6.6\times 10^8~ly$
Work Step by Step
In part (b), we found that $\frac{\Delta \lambda}{\lambda} = \frac{r~\alpha}{c-r~\alpha}$
We can find the distance between the galaxy and the Earth when the light was emitted:
$\frac{\Delta \lambda}{\lambda} = \frac{r~\alpha}{c-r~\alpha} = 0.050$
$r~\alpha = (0.050)(c-r~\alpha)$
$r~\alpha+0.050~r~\alpha = 0.050~c$
$r = \frac{0.050~c}{\alpha(1+0.050)}$
$r = \frac{0.050~c}{1.050~H}$
$r = \frac{(0.050)(3.0\times 10^8~m/s)}{(1.050)~(0.0218~m/s~ly)}$
$r = 6.6\times 10^8~ly$
The distance between the galaxy and the Earth when the light was emitted is $~~6.6\times 10^8~ly$
