Drone Recovery with an Ambiguous Bearing
A survey drone starts at station A. A radio tower T is 6.0 km from A on a bearing of 075°. The drone flies to a point B that is 4.0 km from A and 5.0 km from T. The flight log confirms that B is north of A, meaning its bearing from A lies between 000° and 090°. The operator needs to recover the drone and has two possible bearings in mind from a rough sketch, but only one is consistent with the log. A second drone at A can travel at 30 km/h along a straight line; the operator wants to know the direct distance and bearing to B, and whether a 12-minute flight is enough. Bearings are measured clockwise from north. The operator also asks whether the data would have determined B uniquely without the “north of A” condition, and wants the reason stated in terms of triangle geometry.
1. Using triangle ATB, calculate the angle TAB and list both possible bearings of B from A.
Read the worked answer
In triangle ATB, the known sides are [AT=6,\quad AB=4,\quad BT=5.]
Using the cosine rule at A: [\cos \angle TAB=\frac{AT^2+AB^2-BT^2}{2(AT)(AB)} =\frac{36+16-25}{2(6)(4)}=\frac{27}{48}=0.5625.]
Thus [\angle TAB\approx55.77^\circ] (about 55°46′).
The tower is on bearing 075°, so B can lie on either side of AT: [75^\circ-55.77^\circ\approx19.23^\circ,] [75^\circ+55.77^\circ\approx130.77^\circ.]
So the two possible bearings are approximately 019.2° and 130.8°.
2. Apply the “north of A” condition to choose the bearing, and decide whether a 12-minute flight from A at 30 km/h can reach B.
Read the worked answer
The condition says B has bearing between 000° and 090° from A. Of the two possibilities, only [19.2^\circ] lies in that range. Therefore the correct bearing is about 019.2°.
The direct distance is already given as (AB=4.0) km. At 30 km/h, [t=\frac{4}{30}\ \text{h}=0.1333\ \text{h}=8\ \text{min}.]
A 12-minute flight is enough for a one-way trip from A to B, with about 4 minutes to spare. If the drone had to return to A as well, the round trip would take about 16 minutes and 12 minutes would not be enough.
3. Explain why the original distance data alone do not determine B uniquely, referring to the geometry of the triangle.
Read the worked answer
The three distances (AT=6), (AB=4), and (BT=5) determine the side lengths of triangle ATB, but they do not determine which side of line AT the point B lies on. The two possible positions of B are mirror images across line AT. Both satisfy the same three distance constraints, so both produce congruent triangles and the same angle (\angle TAB).
The bearing is what distinguishes the two mirror-image positions: one gives about 019.2°, the other about 130.8°. The additional “north of A” condition selects the first. This is the geometric ambiguity behind an SSA-style sketch, even though here all three sides are known: side lengths alone fix the triangle up to reflection.