Question

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

Accepted Answer

Draw a diagram to visualize the problem: place points A and B on a horizontal east-west line with A to the west and B to the east, 3.46 miles apart. Mark the bearings from A and B to the transmitter as given. Convert the bearings into angles relative to the east-west line. From A, the bearing 47.7° is measured clockwise from north, so find the angle between the line AB and the line from A to the transmitter. Similarly, convert the bearing from B (302.5°) into an angle relative to the line AB. Use the angles found to determine the directions of the lines from A and B to the transmitter. This will allow you to form a triangle with vertices at A, B, and the transmitter. Apply the Law of Sines to the triangle formed by points A, B, and the transmitter. You know the side AB = 3.46 miles and the two angles at A and B (found from the bearings). Use the Law of Sines formula: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$, where sides are opposite their respective angles. Solve for the side opposite the angle at B, which corresponds to the distance from A to the transmitter. This will give you the required distance without calculating the final numeric value.

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

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Concetti chiave

Bearings and Angle Measurement

Triangle Formation and Law of Sines

Coordinate Geometry and Vector Representation