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The problem asks to find the length of a tunnel through a mountain using trigonometry, based on the given diagram. The following information is provided:

- An angle of \( 83^\circ \).
- Two sides of a triangle, with lengths of 266 meters and 133 meters.
- The task is to find the length of the tunnel (represented as the missing third side).

This is a classic case for applying the **law of cosines**. The law of cosines states:

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

Where:
- \( c \) is the unknown length (the tunnel),
- \( a \) and \( b \) are the two known sides (266 m and 133 m),
- \( C \) is the angle opposite to side \( c \) (83°).

Let's substitute the values:

\[
c^2 = 266^2 + 133^2 - 2 \cdot 266 \cdot 133 \cdot \cos(83^\circ)
\]

To solve this step-by-step:

1. \( 266^2 = 70756 \)
2. \( 133^2 = 17689 \)
3. Calculate \( \cos(83^\circ) \approx 0.12187 \)
4. Now calculate the whole formula:
\[
c^2 = 70756 + 17689 - 2 \cdot 266 \cdot 133 \cdot 0.12187
\]
\[
c^2 = 70756 + 17689 - 8656.65
\]
\[
c^2 = 79788.35
\]
\[
c = \sqrt{79788.35} \approx 282.5 \, \text{meters}
\]

So, the length of the tunnel is approximately **282.5 meters**.

Would you like further details or have any questions?

Here are some related questions you may find interesting:
1. How can the law of cosines be derived from the Pythagorean theorem?
2. What are some real-life applications of the law of cosines besides this tunnel problem?
3. How does the law of sines differ from the law of cosines, and when would you use each?
4. How would the result change if the angle were obtuse or acute?
5. What would happen if one of the side lengths were unknown instead?

**Tip:** When solving problems involving trigonometry, always ensure your calculator is in the correct mode (degrees vs radians) for angle measurements.

Amy wants to know the length of a tunnel built through a mountain using trigonometry. She measures the angle to be 83°, and two sides of a triangle (266 meters and 133 meters) are given. What is the length of the tunnel?

Find the length of a tunnel through a mountain using trigonometry. Given two sides of a triangle (266 m, 133 m) and an angle of 83°, this problem applies the law of cosines to find the missing side. Learn the step-by-step solution for Grades 9-12.

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