A boat heading out to sea starts out at Point *A*, at a horizontal distance of 590 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light from that point to be 11 degrees∘. At some later time, the crew measures the angle of elevation from point B to be 2 degrees∘. Find the distance from point *A* to point *B*. Round your answer to the nearest foot if necessary.

To find the distance from point \( A \) to point \( B \), we need to determine how far the boat has traveled horizontally from point \( A \) to point \( B \).

### Step 1: Calculate the height of the lighthouse beacon light

Given:
- Distance from point \( A \) to the lighthouse (shore), \( d_A = 590 \) feet
- Angle of elevation from point \( A \), \( \theta_A = 11^\circ \)

Using the tangent function:
\[
\tan(\theta_A) = \frac{h}{d_A}
\]
where \( h \) is the height of the lighthouse beacon light. Solving for \( h \):
\[
h = d_A \cdot \tan(\theta_A) = 590 \cdot \tan(11^\circ)
\]

Calculating \( h \):
\[
\tan(11^\circ) \approx 0.1944
\]
\[
h = 590 \cdot 0.1944 \approx 114.696 \text{ feet}
\]

### Step 2: Calculate the distance from point \( B \) to the lighthouse (shore)

Given:
- Angle of elevation from point \( B \), \( \theta_B = 2^\circ \)
- Height of the lighthouse beacon light, \( h = 114.696 \) feet

Using the tangent function again:
\[
\tan(\theta_B) = \frac{h}{d_B}
\]
where \( d_B \) is the distance from point \( B \) to the lighthouse (shore). Solving for \( d_B \):
\[
d_B = \frac{h}{\tan(\theta_B)} = \frac{114.696}{\tan(2^\circ)}
\]

Calculating \( d_B \):
\[
\tan(2^\circ) \approx 0.0349
\]
\[
d_B = \frac{114.696}{0.0349} \approx 3286.43 \text{ feet}
\]

### Step 3: Calculate the horizontal distance from point \( A \) to point \( B \)

The distance from point \( A \) to point \( B \) is the difference between \( d_B \) and \( d_A \):
\[
d_{AB} = d_B - d_A = 3286.43 - 590 \approx 2696.43 \text{ feet}
\]

Rounding to the nearest foot:
\[
d_{AB} \approx 2696 \text{ feet}
\]

Therefore, the distance from point \( A \) to point \( B \) is approximately \( 2696 \) feet.

Would you like more details on any part of the solution? Here are some related questions:

1. How is the tangent function applied in these types of problems?
2. What are other ways to measure the height of a structure using trigonometry?
3. Why is it important to calculate angles of elevation accurately?
4. Can this method be applied to any distance and height measurements?
5. What are the limitations of using tangent functions in real-life scenarios?
6. How do errors in angle measurements affect the final distance calculation?
7. How would this problem change if the initial distance from the lighthouse was different?
8. What other trigonometric functions could be useful in navigation?

**Tip:** When dealing with trigonometric problems, always double-check your angle measurements and ensure your calculator is set to the correct unit (degrees or radians).

Learn how to calculate the distance between two points using the angle of elevation. This trigonometry problem involves measuring angles and applying the tangent function to find distances. Suitable for high school students.

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