Question

A fire is spotted from two lookout stations. The second station located ten miles due east of the first station. The bearing from the first station

to the fire is N 52° E. The bearing from the second station to the fire is N 37° W. To the nearest mile, find the distance from each lookout
station to the fire.

Answer 1

• 8 mi from station 1
• 6 mi from station 2

Explanation:

From the first station, the angle between the direction of the other station (N90E) and the direction of the fire is 90-52 = 38°.

From the second station, the angle between the direction of the other station (N90W) and the direction of the fire is 90-37 = 53°.

We want to find the sides of the triangle with base angles 38° and 53° and base length 10 miles. The a.pex angle will be 180°-38°-53° = 89°. The law of sines can be used.

a/sin(A) = b/sin(B) = c/sin(C)

a = sin(A)·c/sin(C) = sin(38°)·10/sin(89°) ≈ 6.16 ≈ 6 . . . miles (from sta 2)

b = sin(B)·c/sin(C) ≈ sin(53°)·10.0015 ≈ 7.99 ≈ 8 . . . miles (from sta 1)

The fire is about 8 miles from the first station and 6 miles from the second station.