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The problem involves finding the length of a tunnel through a mountain using triangulation, based on two distances and two angles provided by a surveyor. Here's a step-by-step approach to solve this problem using trigonometry:

### Given:
- Distance from one entrance to the surveyor is \( 120 \, \text{m} \).
- Distance from the other entrance to the surveyor is \( 101 \, \text{m} \).
- Angle from one entrance to the perpendicular is \( 42^\circ \).
- Angle from the other entrance to the perpendicular is \( 28^\circ \).

### Step-by-Step Solution:

The tunnel forms a triangle with the surveyor's position and the two entrances of the tunnel. The goal is to find the length of the tunnel, which is the third side of this triangle.

We can use the **Law of Cosines**, which relates the sides and angles of a triangle.

#### Law of Cosines formula:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Where:
- \( c \) is the length of the tunnel (what we're solving for).
- \( a = 120 \, \text{m} \) (distance from the surveyor to one entrance).
- \( b = 101 \, \text{m} \) (distance from the surveyor to the other entrance).
- \( C = 180^\circ - (42^\circ + 28^\circ) = 110^\circ \) (the angle between the two distances).

Now, substituting into the Law of Cosines:

\[
c^2 = 120^2 + 101^2 - 2 \cdot 120 \cdot 101 \cdot \cos(110^\circ)
\]

Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can try solving the equation manually using the Law of Cosines with the formula I provided.

Would you like further assistance with a different question? 

Here are 5 related questions to explore:
1. How is the Law of Cosines different from the Law of Sines in triangle solving?
2. Can you use the Law of Cosines to solve for angles as well as side lengths?
3. What are some real-world applications of trigonometry in surveying?
4. How can trigonometry be applied in architecture or tunnel construction?
5. What are other methods used to measure distances in inaccessible areas?

**Tip**: Always ensure that your calculator is in the correct mode (degrees/radians) when solving trigonometric problems!

A surveyor wants to know the length of a tunnel built through a mountain. According to her equipment, she is located 120 meters from one entrance of the tunnel, at an angle of 42° to the perpendicular. Also according to her equipment, she is 101 meters from the other entrance of the tunnel, at an angle of 28° to the perpendicular. Based on these measurements, find the length of the entire tunnel.

This problem involves finding the length of a tunnel through a mountain by applying the Law of Cosines. Given two distances and two angles from a surveyor's position, learn to calculate the length of the tunnel with detailed trigonometric steps. Suitable for high school students and advanced geometry learners.

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