A surveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 243 meters from one entrance of the tunnel, at an angle of 47° to the perpendicular. Also according to his equipment, he is 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.

To find the length of the tunnel, we can use the Law of Cosines in trigonometry. Let's denote the entrances of the tunnel as points \( A \) and \( B \), and the position of the surveyor as point \( C \). We have the following information:

- \( CA = 243 \) meters
- \( CB = 186 \) meters
- \( \angle ACB = 47° + 27° = 74° \)

We need to find the length of \( AB \), which is the length of the tunnel.

Using the Law of Cosines:
\[ AB^2 = CA^2 + CB^2 - 2 \cdot CA \cdot CB \cdot \cos(\angle ACB) \]

Plugging in the values:
\[ AB^2 = 243^2 + 186^2 - 2 \cdot 243 \cdot 186 \cdot \cos(74°) \]

First, calculate \( 243^2 \):
\[ 243^2 = 59049 \]

Next, calculate \( 186^2 \):
\[ 186^2 = 34596 \]

Then, calculate \( 2 \cdot 243 \cdot 186 \):
\[ 2 \cdot 243 \cdot 186 = 90396 \]

Now, find \( \cos(74°) \):
\[ \cos(74°) \approx 0.2756 \]

Now, substitute these values into the equation:
\[ AB^2 = 59049 + 34596 - 90396 \cdot 0.2756 \]

Calculate the product:
\[ 90396 \cdot 0.2756 \approx 24910.6176 \]

So,
\[ AB^2 = 59049 + 34596 - 24910.6176 \]
\[ AB^2 = 93645 - 24910.6176 \]
\[ AB^2 = 68734.3824 \]

Finally, take the square root to find \( AB \):
\[ AB = \sqrt{68734.3824} \approx 262.11 \text{ meters} \]

Therefore, the length of the tunnel is approximately \( 262.11 \) meters.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. What is the Law of Cosines and how is it used in trigonometry?
2. How can the Law of Sines be applied to solve triangle problems?
3. What are some practical applications of trigonometry in surveying?
4. How do you calculate angles and distances using trigonometric functions?
5. What are other methods to find the length of a side in a triangle given different sets of data?
6. How does changing the angle \( \angle ACB \) affect the length of \( AB \)?
7. Can the length of the tunnel be found using coordinate geometry instead of trigonometry?
8. What are some common errors to avoid when applying the Law of Cosines?

**Tip:** Always double-check your angle measurements and calculations, especially when working with trigonometric functions, as small errors can lead to significant inaccuracies.

Learn how to find the length of a tunnel using trigonometry and the Law of Cosines with a detailed step-by-step solution. Given measurements include distances from surveyor to tunnel entrances and angles. Explore practical applications of trigonometry in surveying. Suitable for high school students.

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