Two fire-lookout stations are 30 miles apart, with station B directly east of station A.

Question

Two fire-lookout stations are 30 miles apart, with station B directly east of station A. Both stations spot a fire on a mountain to the north. The bearing from station A to the fire is 𝑁 28° 𝐸. The bearing from station B is 𝑁 39° 𝑊. How far, to the nearest tenth of a mile, is the fire from station A.

Hello, how do I solve this problem?

Answer 1

Let the fire is located at a point F.

Consider the triangle ABF.

AB = 30 mi

∠B = 39º

∠A = 90º - 28º = 62º

Then ∠F = 180º - (62º + 39º) = 79º

Use the Law of Sines: AF / sin ∠B = AB / sin ∠F

AF / sin 39º = 30 / sin 79º

AF = 30sin 39º/sin 79º ≈ 19.2 mi.