Question

on a map the town of morgan run is due south of davidson and is southeast of vicksburg. The distance from morgan run to davidson and vicksburg are 32 and 52 miles respectively. The distance between davidson and vicksburg is 42 miles. If a plane leaves morgan run to fly to vicksburg, on what bearing should it travel

Answer 1

Put Morgan Run at the origin, with north along +y and east along +x. Let the bearing from Morgan Run to Vicksburg be N θ W (i.e. θ west of north). Then Vicksburg's coordinates are (−52 sin θ, 52 cos θ). The distance from Davidson (0,32) to Vicksburg is 42, so

(52 sin θ)^2 + (52 cos θ − 32)^2 = 42^2.

Simplifying (using sin^2+cos^2=1) gives
2704 − 3328 cos θ + 1024 = 1764,
so cos θ = 1964/3328 = 491/832 ≈ 0.590625.

Thus θ = arccos(491/832) ≈ 53.83°.

So the plane should fly on bearing N 53°50' W, which as an azimuth (clockwise from north) is about 306.17° (≈306.2°).