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To solve this problem, we need to use trigonometric relationships to calculate the total length of the tunnel. The surveyor provides two distances and two angles from either side of the tunnel, relative to a perpendicular line. 

Here's a breakdown:

- One side of the triangle has a distance of **54 meters** with an angle of **56°**.
- The other side has a distance of **31 meters** with an angle of **13°**.
  
We are trying to find the total length of the tunnel, which is the horizontal distance between the two ends.

### Step-by-step Plan:

1. **Determine the horizontal distances for each segment**:
   - Use trigonometry (specifically the cosine function) to find the horizontal distance from the surveyor to each tunnel entrance.
   - For the left side, we use the angle of **56°**:
     \[
     d_1 = 54 \times \cos(56^\circ)
     \]
   - For the right side, we use the angle of **13°**:
     \[
     d_2 = 31 \times \cos(13^\circ)
     \]

2. **Add the two horizontal distances to find the total length of the tunnel**:
   \[
   \text{Total length} = d_1 + d_2
   \]

Let me calculate this now.The total length of the tunnel is approximately **60.4 meters**, rounded to the nearest tenth.

Would you like further details on any part of the solution, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How would the solution change if the angles were different?
2. What if the distances from the two ends of the tunnel were different?
3. Can you apply the sine rule to solve similar problems involving non-right triangles?
4. How does the cosine of an angle relate to horizontal distances in trigonometry?
5. How could you calculate the height of the mountain above the surveyor based on this setup?

**Tip:** Always ensure angles are in the correct units (degrees or radians) when using trigonometric functions in calculations!

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

Learn how to calculate the length of a tunnel built through a mountain using trigonometry. The surveyor measures distances of 54 meters and 31 meters from two entrances, at angles of 56° and 13° to the perpendicular, respectively. This problem covers cosine function applications in trigonometry.

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