A boat is heading towards a lighthouse, where Jaxson is watching from a vertical distance of 103 feet above the water. Jaxson measures an angle of depression to the boat at point AA to be 12degrees 
∘
 . At some later time, Jaxson takes another measurement and finds the angle of depression to the boat (now at point BB) to be 64degrees 
∘
 . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

To solve this problem, we can break it down using trigonometry, particularly by employing right triangle properties. Here's how to approach it:

### Step 1: Identify the problem setup.
- Jaxson is at the top of a lighthouse, 103 feet above the water.
- The angles of depression to the boat at two points (AA and BB) are 12° and 64°, respectively.
- We are asked to find the horizontal distance from point AA to point BB.

### Step 2: Use trigonometric relationships.
We will treat this as two right triangles:
- In both cases, the height of the lighthouse (opposite side) is 103 feet.
- The horizontal distance to points AA and BB (adjacent side) can be found using the tangent of the angles of depression.

Let:
- \( d_A \) be the horizontal distance from the lighthouse to point AA.
- \( d_B \) be the horizontal distance from the lighthouse to point BB.
  
The tangent function relates the angle of depression to the horizontal distance:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- The opposite side is the height of the lighthouse (103 feet).
- The adjacent side is the horizontal distance.

For point AA (\(\theta = 12^\circ\)):
\[
\tan(12^\circ) = \frac{103}{d_A}
\]
Solving for \(d_A\):
\[
d_A = \frac{103}{\tan(12^\circ)}
\]

For point BB (\(\theta = 64^\circ\)):
\[
\tan(64^\circ) = \frac{103}{d_B}
\]
Solving for \(d_B\):
\[
d_B = \frac{103}{\tan(64^\circ)}
\]

### Step 3: Find the distance from point AA to point BB.
The distance between AA and BB is simply the difference between \(d_A\) and \(d_B\):
\[
\text{Distance} = d_A - d_B
\]

Let's calculate this now.

### Step 4: Perform the calculations.The horizontal distances are as follows:
- The distance from the lighthouse to point AA (\(d_A\)) is approximately 485 feet.
- The distance from the lighthouse to point BB (\(d_B\)) is approximately 50 feet.

Therefore, the distance between point AA and point BB is approximately **434 feet**.

Would you like further details or explanations? Here are five related questions you could explore:
1. How would the solution change if the lighthouse were taller?
2. Can you find the speed of the boat if the time between the two measurements is known?
3. How would the calculation change if the angle of elevation was given instead of depression?
4. What is the significance of the tangent function in trigonometry?
5. How do you solve similar problems involving angles of elevation?

**Tip:** Always ensure you're working in consistent units and double-check angle conversions when dealing with trigonometric functions.

A boat is heading towards a lighthouse, where Jaxson is watching from a vertical distance of 103 feet above the water. Jaxson measures an angle of depression to the boat at point AA to be 12°. At some later time, Jaxson takes another measurement and finds the angle of depression to the boat (now at point BB) to be 64°. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Solve a problem involving a lighthouse and a boat where the angles of depression to two points are given. This problem explores trigonometric concepts and formulas, such as the tangent function, to find the horizontal distance between two points. Suitable for students in Grades 10-12.

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