Two firelookout stations are 300 miles apart with station A

Two fire-lookout stations are 300 miles apart, with station A directly south of station B. Both stations spot a fire. The bearing of the fire from station A is N55 degree E and the bearing of the fire from station B is S60 degree E. How far, to the nearest tenth of a mile, is the fire from each lookout station? The distance from station B to the fire is miles (Round to the nearest tenth.)

Solution

Here we have that the sum of all three angles of a triangle is 180 degree, so

missing angle at fire point =180-(60+55) =180-115= 65 degree

Now on applying sine rule in this triangle, we have

a/sin A = c/sin C

or a/sin 55 = 300/sin 65

or a/0.819 = 300/0.906

a= 300(0.819)/0.906 = 245.73/0.906 =271.1

so required distance is 271.1 miles.

Answer