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This problem involves applying the Law of Cosines to a triangle formed by the two measurements (243 meters and 186 meters) and the angles given (47° and 27°). The angle between these two measurements would be the supplementary angle of 47° and 27°, which is \( 180^\circ - 47^\circ - 27^\circ = 106^\circ \).

We can solve for the total length of the tunnel using the Law of Cosines, which is formulated as:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( c \) is the length of the tunnel (the side opposite the angle),
- \( a = 243 \) meters,
- \( b = 186 \) meters,
- \( C = 106^\circ \).

Let's calculate that.The length of the entire tunnel is approximately 344.3 meters.

Do you need further details or explanations? Here are five related questions to expand on this:

1. How would the problem change if the angles were different?
2. Can you solve this using a different trigonometric approach, like the Law of Sines?
3. How does the Law of Cosines differ from the Pythagorean theorem?
4. What happens if you have a right triangle instead of an oblique one in this scenario?
5. Why do we need to convert angles to radians when using trigonometric functions?

**Tip:** Always remember to check if your triangle is a right triangle first—if it is, you can use simpler trigonometric methods like sine, cosine, or the Pythagorean theorem!

A surveyor wants to know the length of a tunnel built through a mountain. According to her equipment, she is located 243 meters from one entrance of the tunnel, at an angle of 47° to the perpendicular. She is also 186 meters from the other entrance of the tunnel, at an angle of 27° to the perpendicular. Based on these measurements, find the length of the entire tunnel.

Solve a word problem involving trigonometry by calculating the length of a tunnel using the Law of Cosines. The problem includes angles of 47° and 27°, distances of 243 meters and 186 meters, and a solution for the total tunnel length. Ideal for high school students (Grades 10-12).

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