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This problem can be solved using the Law of Sines, since we are dealing with a triangle formed by the surveyor’s position, the two ends of the tunnel, and the angles given.

### Step-by-step solution:

1. **Label the triangle**: 
   - \( A \) is the vertex where the surveyor is located.
   - \( B \) and \( C \) are the two tunnel ends.
   - \( AB = 54 \) meters, \( AC = 31 \) meters.
   - The angles at \( A \) are \( 56^\circ \) and \( 13^\circ \).

2. **Find the angle at \( A \)**:
   The sum of the interior angles of a triangle is \( 180^\circ \). So the angle at \( A \) is:
   \[
   \angle A = 180^\circ - 56^\circ - 13^\circ = 111^\circ
   \]

3. **Apply the Law of Sines**:
   The Law of Sines states:
   \[
   \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
   \]
   Here:
   - \( a \) is the unknown length of the tunnel (which is \( BC \)).
   - \( b = 31 \) meters, \( \angle B = 56^\circ \)
   - \( c = 54 \) meters, \( \angle C = 13^\circ \)
   
   Using the Law of Sines, we can set up the following:
   \[
   \frac{BC}{\sin(111^\circ)} = \frac{31}{\sin(56^\circ)}
   \]
   
4. **Solve for \( BC \)**:
   Rearranging the equation:
   \[
   BC = \frac{31 \times \sin(111^\circ)}{\sin(56^\circ)}
   \]

Let me calculate the result.The length of the tunnel is approximately \( 34.9 \) meters.

Would you like more details or have any further questions?

Here are some related questions for practice:
1. How would the solution change if the angles were different?
2. What if the distance from the surveyor to one end of the tunnel were unknown, how would you solve that?
3. Can the Law of Cosines be applied to this problem?
4. What if the surveyor made a mistake in measuring the angles by a few degrees—how would that affect the result?
5. How does rounding affect the precision of the solution?

**Tip**: The Law of Sines is often useful in non-right triangles when you know two angles and one side or two sides and a non-included angle.

A surveyor wants to know the length of a tunnel built through a mountain. She is located 54 meters from one entrance, at an angle of 56° to the perpendicular, and 31 meters from the other entrance, at an angle of 13° to the perpendicular. Based on these measurements, find the length of the entire tunnel.

This problem uses the Law of Sines to find the length of a tunnel. Given the surveyor's angles (56° and 13°) and distances (54 meters and 31 meters) from two ends of the tunnel, this step-by-step solution demonstrates how to apply trigonometry to solve the problem.

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