From the top of the mountain 532 m higher than a nearby river, the angle of depression of a point P on the closer bank of the river is 52.6o, and the angle of depression of a point Q directly opposite P on the other side is 34.5o. Points P and Q and the foot of the mountain are on the same horizontal line. Find the distance across the river from P to Q.
The distance, M, from the base of the mountain to the nearest bank can be found as
tan 52.6 = 532/ M
Rearrange as
M = 532 / tan 52.6 ≈ 406.7 ft
The distance,N, from the base of the mountain to the far bank can be found as
tan 34.5 = 532/ N
N = 532 / tan 34.5 ≈ 774.1 ft
The width of the river = N - M = 774.1 - 406.7 ≈ 367.4 ft