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.