Do the Math please, How to use the Law of Sines to Locate a Fire...??

Question

The bearing from the Pine knob fire tower to the Colt Station fire tower is N 65° E, and the two towers are 30 kilometers apart. A fire spotted by rangers in each tower has a bearing of N 80° E from Pine Knob and S 70° E from Colt Station. Find the distance of the fire from each tower.

Answer 1

A=30°, a=30 km
B=135°, b=distance from Pine Knob to fire
C=15°, c=distance from Colt station fire tower to fire

Apply the sine rule:

a / Sin A = b / Sin B = c / Sin C

30 / Sin 30° = b / Sin 135° = c / Sin 15­
=>
b = distance from Pine Knob to fire = 42.43 km
c = distance from Colt station fire tower to fire = 15.53 km

Answer 2

If you draw a figure, the triangle formed by P, C, and the fire F has a (65+70)= 135 degree angle at C, a (25-10) = 15 degree angle at B, leaving 30 degrees for the angle at F.

so sin 30 / 30 = sin 135/ [fire to P] = sin 15/ [fire to C]

Answer 3

sin(30)/30=sin(15)/x so x=30sin(15)/sin(30)=15.53 miles from colt station.
sin(30)/30=sin(135)/y or y=30*sin(135)/sin(30)=10.6miles from Pine Knob.