A boat heading out to sea starts out at Point A*A*, at a horizontal distance of 853 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 7degrees∘. At some later time, the crew measures the angle of elevation from point B*B* to be 4degrees∘. Find the distance from point A*A* to point B*B*. Round your answer to the nearest foot if necessary.

To solve the problem, we can use trigonometry, specifically the tangent function, as the problem involves angles of elevation and distances.

### Problem Restatement:
- **Point A:** The boat is 853 feet from the lighthouse, and the angle of elevation to the lighthouse beacon is 7°.
- **Point B:** The angle of elevation from this point is 4°.
- **Goal:** Find the distance between Point A and Point B.

### Step-by-Step Solution:

1. **Height of the Lighthouse (h):**
   - Using the information at Point A, where the horizontal distance is 853 feet and the angle of elevation is 7°:
     \[
     \tan(7^\circ) = \frac{h}{853}
     \]
     Solving for \( h \):
     \[
     h = 853 \times \tan(7^\circ)
     \]

2. **Distance from Point B to the Lighthouse:**
   - Let's denote the distance from Point B to the lighthouse as \( x \).
   - The angle of elevation at Point B is 4°, so:
     \[
     \tan(4^\circ) = \frac{h}{x}
     \]
     Solving for \( x \):
     \[
     x = \frac{h}{\tan(4^\circ)}
     \]

3. **Distance between Point A and Point B:**
   - The distance between Point A and Point B is the difference between the horizontal distances from each point to the lighthouse:
     \[
     \text{Distance AB} = x - 853
     \]

Now, let's calculate these values.The distance between Point A and Point B is approximately **645 feet**.

Would you like further explanation on any part of the solution, or do you have any additional questions?

Here are 5 related questions you might find interesting:
1. How would the distance between the points change if the initial distance from the lighthouse was greater than 853 feet?
2. What if the angles of elevation were closer in value, such as 6° and 5°? How would that affect the distance between Point A and Point B?
3. Can we determine the exact position of the boat relative to the lighthouse at any given time using similar methods?
4. How can you find the height of the lighthouse if only the distance and one angle of elevation are given?
5. How would the problem change if the angle of depression was given instead of the angle of elevation?

**Tip:** When working with trigonometric problems involving angles of elevation or depression, drawing a diagram can help visualize the problem and ensure that the correct relationships are used.

Learn how to calculate the distance between two points using trigonometry with 7° and 4° angles of elevation. This problem involves applying the tangent function to find distances based on angles of elevation.

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