A forest ranger at an observation point A sights a fire in t

A forest ranger at an observation point A sights a fire in the direction N27°10\'E. Another ranger at an observation point B, 5.0 miles due east of A, sights the same fire at N52°40\'W. Approximate the distance from each of the observation points to the fire. (Round your answers to two decimal places.)

Solution

Let the fire be at point C

In triangle ABC,

angle A = 90° - (27° 10)\' = 62° 50\'
angle B = 90° - (52° 40\') = 37° 20\'
AB = 5.0

angle C = 180° - (62° 50\') - (37° 20\') = 79° 50\'

(AC)/sinB = (AB)/sinC
AC = (AB)sinB / sinC
= (5.0)sin(37° 20\') / sin(79° 50\')
3.08 miles

(BC)/sinA = (AB)/sinC
BC = (AB)sinA / sinC
= (6.0)sin(62° 50\') / sin(79° 50\')
4.5 miles