A boat is heading towards a lighthouse, whose beacon-light is 104 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 11degrees 
∘
 , before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 22degrees 
∘
 . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.


To solve this problem, we can use trigonometry to determine the distance between points \( A \) and \( B \). Here’s a step-by-step approach:

### Given:
- Height of the lighthouse (\( h \)) = 104 feet.
- Angle of elevation from point \( A \) (\( \theta_1 \)) = 11°.
- Angle of elevation from point \( B \) (\( \theta_2 \)) = 22°.

### To Find:
- The distance between points \( A \) and \( B \).

### Approach:
1. **Define the distances**:
   - Let \( x \) be the distance from point \( A \) to the base of the lighthouse.
   - Let \( y \) be the distance from point \( B \) to the base of the lighthouse.
   - The distance between \( A \) and \( B \) is \( x - y \).

2. **Use trigonometric relationships**:
   From point \( A \), using the tangent of angle \( \theta_1 \):
   \[
   \tan(11^\circ) = \frac{h}{x}
   \]
   So, \( x = \frac{104}{\tan(11^\circ)} \).

   From point \( B \), using the tangent of angle \( \theta_2 \):
   \[
   \tan(22^\circ) = \frac{h}{y}
   \]
   So, \( y = \frac{104}{\tan(22^\circ)} \).

3. **Find the distance between \( A \) and \( B \)**:
   \[
   \text{Distance between } A \text{ and } B = x - y = \frac{104}{\tan(11^\circ)} - \frac{104}{\tan(22^\circ)}
   \]

### Calculation:
Now, let's calculate this using a calculator for the tangent values:

- \(\tan(11^\circ) \approx 0.1944\)
- \(\tan(22^\circ) \approx 0.4040\)

\[
x = \frac{104}{0.1944} \approx 535.12 \text{ feet}
\]
\[
y = \frac{104}{0.4040} \approx 257.43 \text{ feet}
\]

\[
\text{Distance between } A \text{ and } B = 535.12 - 257.43 \approx 278 \text{ feet}
\]

### Final Answer:
The distance from point \( A \) to point \( B \) is approximately **278 feet**.

Would you like more details or have any questions about this solution?

### Related Questions:
1. How would the answer change if the height of the lighthouse was 200 feet instead?
2. What is the significance of the angle of elevation in trigonometric problems?
3. How can we verify this solution using a different trigonometric function?
4. What is the formula for tangent and how is it used in right triangle problems?
5. If the boat continued to move, what would the angle of elevation be at a distance of 100 feet from point \( B \)?

### Tip:
In problems involving right triangles and angles of elevation, using tangent is often helpful since it relates the height of an object to the distance from the object.

A boat is heading towards a lighthouse, whose beacon-light is 104 feet above the water. From point A, the boat’s crew measures the angle of elevation to the beacon, 11°, before they draw closer. They measure the angle of elevation a second time from point B at some later time to be 22°. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

Solve a trigonometry problem involving angles of elevation from two points to a 104-foot lighthouse. Learn how to calculate the distance between the points using tangent functions. Suitable for students in Grades 9-12.

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