We need to find the distances from two towers (A and B) to a fire. The directions from the towers to the fire are given, and the distance between the towers is known.
- Tower A spots the fire at a direction of 347°.
- Tower B spots the fire at a direction of 59°.
- The distance between Tower A and Tower B is 40 miles.
- The angle between the directions from the towers to the fire is 317° - 59° = 258°.
We can use the Law of Sines to find the distances from the towers to the fire. The Law of Sines states:
sin(A)a=sin(B)b=sin(C)c
where a, b, and c are the sides of the triangle, and A, B, and C are the opposite angles.
- Angle at Tower A (A) = 360° - 347° = 13°
- Angle at Tower B (B) = 59°
- Angle at the fire (C) = 180° - (13° + 59°) = 108°
Using the Law of Sines:
sin(C)AB=sin(A)a=sin(B)b
where AB=40 miles, C=108°, A=13°, and B=59°.
sin(108°)40=sin(13°)a=sin(59°)b
Solving for a (distance from Tower A to the fire):
a=sin(108°)40⋅sin(13°)
Solving for b (distance from Tower B to the fire):
b=sin(108°)40⋅sin(59°)
- Distance from Tower A to the fire: a≈9 miles (rounded to the nearest whole number)
- Distance from Tower B to the fire: b≈35 miles (rounded to the nearest whole number)