Travelling along a straight line at a speed of 170 m s^ -1 , a...

The apparent frequency registered by the detector is given by the Doppler effect formula:

$f = f_0 \left( \dfrac{v}{v - v_s \cos\theta} \right)<br><br>where$v$is the velocity of sound,$v_s$is the velocity of the source, and \theta is the angle between the source's velocity vector and the line joining the source to the detector at the instant the sound is emitted.<br><br>Given that the emitted frequency$f_0 = 1200 $\text{ Hz} and the registered frequency$f = 1200 $\text{ Hz}, we have:<br><br>$1200 = 1200 \left( \dfrac{v}{v - v_s \cos\theta} \right)$<br><br>$\Rightarrow v - v_s \cos\theta = v$<br><br>$\Rightarrow \cos\theta = 0 \Rightarrow \theta = 90^{{}^{{}^{\circ}}}$<br><br>This means the sound heard by the detector was emitted when the source was at the point of closest approach. At this instant, the distance between the source and the detector is$d = 200 \text{ m}$.<br><br>The time taken for this sound to reach the detector is:<br><br>$t = \dfrac{d}{v} = \dfrac{200}{340} = \dfrac{10}{17} \text{ s}$<br><br>During this time, the source continues to move forward. The distance traveled by the source in time$t$is:<br><br>$x = v_s t = 170 \times \dfrac{10}{17} = 100 \text{ m}$<br><br>At the instant the detector registers the frequency, the source has moved$100 \text{ m}$from the point of closest approach. The distance$D$separating the source and the detector at this instant is:<br><br>$D = \sqrt{x^2 + d^2} = \sqrt{100^2 + 200^2}$<br><br>$D = \sqrt{10000 + 40000} = \sqrt{50000} = 100\sqrt{5} \text{ m}$<br><br>Using$\sqrt{5} \approx 2.236$, we get:<br><br>$D \approx 100 \times 2.236 = 223.6 \text{ m}$<br><br>Rounding to the nearest integer, the distance is$224 \text{ m}$.$