To solve this problem, we’ll use trigonometry, specifically the Law of Sines, as we have an oblique triangle with angles and a side. Here’s the step-by-step solution:

Step 1: Define the known points and directions.

1. Tower A (let’s call it point $A$) spots the fire in the direction of 324°.
2. Tower B (let’s call it point $B$) is located 60 miles away from Tower A in the direction of 40° from Tower A.
3. Tower B spots the fire in the direction of 323°.

Let’s place Tower A at the origin and use the directions to form a triangle.

Step 2: Determine the angle between the sightlines of A and B.

1. Angle at point A (angle $\angle FAB$): Since the fire is at 324° and tower B’s direction from A is 40°, we find the difference: $324^\circ - 40^\circ = 284^\circ$ So, $\angle FAB = 284^\circ$.

2. Angle at point B (angle $\angle FBA$): Similarly, the angle between the fire’s direction (323°) and the position of Tower A (40°) as seen from B is: $323^\circ - 40^\circ = 283^\circ$ So, $\angle FBA = 283^\circ$.

Since both angles are essentially supplementary, meaning they add up to approximately 180°, we’re able to solve this as an approximately linear problem.

Step 3: