a ranger in tower A spots a fire at a direction of 354degrees. a ranger in tower B, located 60miles a direction of 42degrees from tower A, spots the fire at a direction of 320degrees. how far from tower A is the fire?

To solve this problem, let's break down the information given and apply trigonometry, specifically the Law of Sines or Cosines, to find the distance from Tower A to the fire. Here’s the information we have:

1. **Position of Towers A and B**: Tower B is located 60 miles away from Tower A, in a direction of 42 degrees from Tower A.
2. **Angles from Towers to the Fire**:
   - Tower A spots the fire at a direction of 354 degrees.
   - Tower B spots the fire at a direction of 320 degrees.

### Step 1: Setting up a Triangle with the Given Information

To simplify, let’s define:
- \( A \) as the position of Tower A,
- \( B \) as the position of Tower B, and
- \( F \) as the location of the fire.

We need to determine the angle at each point in the triangle \( \triangle ABF \).

### Step 2: Determining Angles in Triangle \( \triangle ABF \)

1. **Angle \( \angle A \)**:
   - The direction of the fire from Tower A is 354 degrees, which is 6 degrees west of true north.
   - Since Tower B is located at a 42-degree angle from Tower A, the interior angle \( \angle A \) in \( \triangle ABF \) is the difference between these two directions: 
     \[
     \angle A = 42^\circ - 6^\circ = 36^\circ.
     \]

2. **Angle \( \angle B \)**:
   - The direction of the fire from Tower B is 320 degrees, which is 40 degrees west of north.
   - The relative position of Tower A from Tower B is at a 42-degree angle, so the interior angle \( \angle B \) in \( \triangle ABF \) is:
     \[
     \angle B = 320^\circ - 42^\circ = 278^\circ.


A ranger in tower A spots a fire at a direction of 354 degrees. A ranger in tower B, located 60 miles in a direction of 42 degrees from tower A, spots the fire at a direction of 320 degrees. How far from tower A is the fire?

Discover how to determine the distance from Tower A to a fire based on directional observations from rangers in two towers. This solution applies trigonometric principles, including the Law of Sines and Law of Cosines, and is ideal for high school students (Grades 10-12).

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