To solve this problem, we need to find the distances from tower A and tower B to the fire using trigonometry, specifically the Law of Sines or Law of Cosines.

Step 1: Analyze the Triangle Formed

The points in the diagram form a triangle:

• Let $A$ be the position of tower A,
• $B$ be the position of tower B, which is 60 miles away from A,
• $C$ be the position of the fire.

The angles provided are:

• $\angle CAB = 40^\circ$, which is the angle between the line from A to B and the line from A to C.
• The direction angles help determine the interior angles of the triangle:
  • From the directions given, we can find the angle $\angle ABC$ as $323^\circ - 324^\circ = 1^\circ$.
  • The angle $\angle ACB$ would then be $180^\circ - (40^\circ + 1^\circ) = 139^\circ$.

Step 2: Use the Law of Sines

With angle $\angle ACB = 139^\circ$, side $AB = 60$ miles, and angle $\angle CAB = 40^\circ$, we can apply the Law of Sines:

$\frac{AC}{\sin(1^\circ)} = \frac{60}{\sin(139^\circ)}$

Solving this equation for $AC$ will give the distance from tower A to the fire.

Let me calculate that.The fire is approximately 2 miles from tower A.

Would you like further details or have any questions?

Here are some related questions:

1. How can we find the distance from the fire to tower B?
2. What if the directions were different; how would that change the calculations?
3. How is the Law of Sines applied in non-right triangles?
4. Why is it important to round to whole numbers in distance calculations?
5. Can the Law of Cosines be used here instead of the Law of Sines?

Tip: In trigonometric problems, always double-check angle measures, as they are crucial for accurate results.