Question

ship a is 7 miles from a lighthouse on the shore. its bearing from the lighthouse is s 15 degrees w. ship b is 12 miles from the same lighthouse. its bearing from the lighthouse is s 82 degrees w. find the number of miles gbetween the two ships

Answer 1

Place the lighthouse at the origin. The bearings S 15° W and S 82° W correspond to central angles 195° and 262° from north, so the angle between the two position vectors is 262° − 195° = 67°.

Use the law of cosines in the triangle formed by the lighthouse and the two ships:
g^2 = 7^2 + 12^2 − 2·7·12·cos(67°) = 49 + 144 − 168·cos(67°).

Compute:
cos(67°) ≈ 0.390731,
g^2 ≈ 193 − 168·0.390731 ≈ 127.357,
g ≈ √127.357 ≈ 11.29 miles.

So the distance between the ships is about 11.29 miles.