A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of 138 feet above the water. Dalvin measures an angle of depression to the boat at point AA to be 13degrees 
∘
 . At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point BB) to be 45degrees 
∘
 . Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

This problem involves using trigonometry to find the distance between two points along the water (points AA and BB) using the angles of depression measured by Dalvin from the top of the lighthouse.

### Key Information:
- Height of Dalvin above the water = 138 feet
- Angle of depression to point AA = 13°
- Angle of depression to point BB = 45°

The distance we are asked to find is the horizontal distance between points AA and BB.

### Steps:

1. **Use trigonometry to find the horizontal distance to point AA**:
   - The angle of depression forms a right triangle with the height of the lighthouse and the horizontal distance.
   - For point AA, we have:
     \[
     \tan(13^\circ) = \frac{\text{height of lighthouse}}{\text{horizontal distance to AA}} = \frac{138}{d_{AA}}
     \]
     Solving for \( d_{AA} \):
     \[
     d_{AA} = \frac{138}{\tan(13^\circ)} 
     \]

2. **Use trigonometry to find the horizontal distance to point BB**:
   - For point BB, we have:
     \[
     \tan(45^\circ) = \frac{138}{d_{BB}} 
     \]
     Since \( \tan(45^\circ) = 1 \), the horizontal distance to point BB is:
     \[
     d_{BB} = 138 \text{ feet}
     \]

3. **Find the distance between points AA and BB**:
   - The distance between points AA and BB is the difference between the two horizontal distances:
     \[
     \text{Distance from AA to BB} = d_{AA} - d_{BB}
     \]

Now, let's calculate this:

### Calculations:
1. \( d_{AA} = \frac{138}{\tan(13^\circ)} \approx \frac{138}{0.2315} \approx 596.1 \text{ feet} \)
2. \( d_{BB} = 138 \text{ feet} \)

So, the distance from AA to BB is:
\[
\text{Distance from AA to BB} = 596.1 - 138 = 458.1 \text{ feet}
\]

### Final Answer:
The distance from point AA to point BB is approximately **458.1 feet**.

Let me know if you would like more details, or if you have any other questions!

Here are 5 relative questions:
1. How does the angle of depression relate to the triangle in this scenario?
2. Why is the tangent function used to calculate horizontal distances?
3. What happens if the angle of depression at point BB was greater than 45°?
4. How would the result change if the height of the lighthouse was doubled?
5. How could you find the total distance the boat traveled if it made several stops at different points?

**Tip**: The tangent of an angle in a right triangle relates the opposite side to the adjacent side, which makes it very useful for solving real-world problems involving angles of elevation or depression.

A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of 138 feet above the water. Dalvin measures an angle of depression to the boat at point AA to be 13°. At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point BB) to be 45°. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

This problem explores how to find the distance between two points using trigonometry and angles of depression from a lighthouse. Given a height of 138 feet and angles of 13° and 45°, learn how to calculate the horizontal distances and solve for the distance between points AA and BB. Suitable for students in Grades 9-12.

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