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To solve this problem, we need to find the horizontal distance \( x \) from the boat \( B \) to the base of the lighthouse \( L \), using the given information. Let's analyze the diagram and the data provided:

- The height of the lighthouse is \( 215 \) feet (from point \( L \) to point \( T \)).
- The angle of elevation from the boat to the top of the lighthouse is \( 26^\circ \).
- The horizontal distance from the boat to the lighthouse's base is labeled \( x \).

This forms a right triangle with:
- The height of the lighthouse (\( 215 \) feet) as the opposite side.
- The horizontal distance (\( x \)) as the adjacent side.
- The angle of elevation \( 26^\circ \).

### Using Trigonometry:
We can use the tangent function, which relates the opposite and adjacent sides of a right triangle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Substituting the given values:
\[
\tan(26^\circ) = \frac{215}{x}
\]
Solving for \( x \):
\[
x = \frac{215}{\tan(26^\circ)}
\]

Let me calculate the value of \( x \).The horizontal distance \( x \) from the boat to the base of the lighthouse is approximately \( 440.82 \) feet.

Would you like more details on this or have any other questions?

Here are five related questions you might find useful:
1. How do we determine when to use tangent, sine, or cosine in right triangle problems?
2. What happens if the angle of elevation changes? How would that affect the horizontal distance?
3. Can we calculate the distance if we only know the angle and the horizontal distance?
4. What is the relationship between angle size and the horizontal distance in a right triangle?
5. How can we use this method to calculate distances in real-world navigation scenarios?

**Tip:** When working with right triangles, remember that tangent relates the opposite and adjacent sides, which is useful when dealing with elevation angles and heights.

The top of a lighthouse, T, is 215 feet above sea level, L, against a wall and makes an angle of 26° with the boat. Find the horizontal distance, x, from the boat to the base of the lighthouse.

Learn how to use the tangent function to calculate the horizontal distance from a boat to the base of a lighthouse. Given the lighthouse height of 215 feet and an elevation angle of 26°, this example demonstrates a practical application of right-triangle trigonometry.

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