Answer
The distance between the source and the detector is $80~km$
Work Step by Step
We can use the speed $v_1$ and the travel time $t_1$ of the longitudinal waves to write an expression for the distance:
$d = v_1~t_1$
Note that $t_2 = t_1+2.0~s$. We can use the speed $v_2$ and the travel time $t_2$ of the transverse waves to write an expression for the distance:
$d = v_2~t_2 = v_2~(t_1+2.0~s)$
Since the distance is the same, we can equate the two equations to find $t_1$:
$v_1~t_1 = v_2~(t_1+2.0~s)$
$(v_1-v_2)~t_1 = (2.0~s)~v_2$
$t_1 = \frac{(2.0~s)~v_2}{v_1-v_2}$
$t_1 = \frac{(2.0~s)(8.0~km/s)}{10.0~km/s-8.0~km/s}$
$t_1 = 8.0~s$
We can find the distance $d$:
$d = v_1~t_1 = (10.0~km/s)(8.0~s) = 80~km$
The distance between the source and the detector is $80~km$.
