Answer
(a) The x-coordinate of the source is -2.0 meters
(b) The y-coordinate of the third listener is 4.6 meters.
Work Step by Step
(a) If the same wave front reaches two listeners, the two listeners must be the same distance from the source. We can find the x-coordinate $r$ of the source.
$r = \frac{r_1+r_2}{2}$
$r = \frac{-7.0~m+3.0~m}{2}$
$r = -2.0~m$
The x-coordinate of the source is -2.0 meters.
(b) If the same wave front reaches a third listener at the same time, the third listener must be the same distance from the source. Since the first two listeners are a distance of 5.0 meters from the source, the listener on the positive y-axis must be a distance of 5.0 meters from the source. We can find the y-coordinate of this listener.
$\sqrt{r^2+y^2} = 5.0~m$
$r^2+y^2 = 25.0~m^2$
$y = \sqrt{25.0~m^2- 4.0~m^2}$
$y = 4.6~m$
The y-coordinate of the third listener is 4.6 meters.
