i can't solve this word problem

Question

A ranger in tower A spots a fire at a direction of 320º. A ranger in tower B, located 60 mi at a direction of 59° from tower A, spots the fire at a direction of 290°. How far from tower a is the fire? How far from tower B?

 

Tower A from fire ______________mi

 

Tower B from fire______________mi

Answer 1

To solve this problem, start by drawing a diagram.  Recognize that degree bearings are relative to due north.  Start by labeling a point A.  Draw a vertical line (north-south) through this point.  Draw a line from point A at an angle 59 degrees down from the vertical line (heading approximately east northeast of A).  label a point B on this line that will represent a distance of 60 miles from point A.  Draw a line from point A, heading just north of northwest, to the point of the fire (label is as C).  The line from point A to point C will have an angle 40 degrees down from the north line that goes through point A.  Draw a line from point B to point C.  Now you should have a triangle ABC.  Draw a vertical line through point B (north-south).  The line from point B to point C should be 70 degrees down (to the approximate west northwest of B) from the vertical line through point B.  Angle A has a measure of 40 + 59 = 99 degrees.  Angle B has a measure of 180 - 70 - 59 = 51 degrees.  Angle C can be found by subtracting angle A and angle B from 180 = 30 degrees.  Now use the law of sines to find the lengths of AC and BC:

60/sin(30) = AC/sin(51) = BC/sin(99)  -->  AC = 93.26 mi     BC = 118.52 mi