A boat heading out to sea starts out at Point AA, at a horizontal distance of 1032 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 15degrees 
∘
 . At some later time, the crew measures the angle of elevation from point BB to be 6degrees 
∘
 . Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

To solve this problem, we can use trigonometry, particularly the tangent function, which relates the angle of elevation, the height of the lighthouse, and the horizontal distance from the lighthouse.

### Step 1: Defining variables and concepts
- Let the height of the lighthouse be \( h \) (in feet).
- Point AA is 1032 feet away from the lighthouse horizontally.
- At Point AA, the angle of elevation is \( 15^\circ \).
- At Point BB, the angle of elevation is \( 6^\circ \).
- We are tasked with finding the distance from Point AA to Point BB, which we’ll call \( d \) (the distance traveled by the boat).

### Step 2: Using the tangent function
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

For Point AA (where the angle of elevation is \( 15^\circ \)):
\[
\tan(15^\circ) = \frac{h}{1032}
\]
From this, we can solve for the height of the lighthouse \( h \):

\[
h = 1032 \times \tan(15^\circ)
\]

### Step 3: Setting up the equation for Point BB
Let the horizontal distance from the lighthouse to Point BB be \( 1032 + d \). At Point BB, the angle of elevation is \( 6^\circ \), so we apply the tangent function again:

\[
\tan(6^\circ) = \frac{h}{1032 + d}
\]
Substitute \( h \) from the earlier equation into this expression:

\[
\tan(6^\circ) = \frac{1032 \times \tan(15^\circ)}{1032 + d}
\]

### Step 4: Solving for \( d \)
Now we can solve this equation to find \( d \), the distance from Point AA to Point BB.

Let me compute the result.The distance from Point AA to Point BB is approximately 1598.9 feet.

Would you like any further details or explanations on this solution?

Here are 5 related questions that extend this topic:
1. How would the solution change if the boat started closer to the lighthouse?
2. Can you derive a general formula for the distance between two points with known angles of elevation?
3. How would you apply this to a real-world scenario with elevation errors?
4. What would happen if the angles of elevation were measured in radians instead of degrees?
5. How would the result change if the lighthouse's height was directly given?

**Tip:** Always double-check your angle units when working with trigonometric functions. Most calculators default to radians, so switching to degrees might be necessary.

A boat heading out to sea starts out at Point AA, at a horizontal distance of 1032 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 15 degrees. At some later time, the crew measures the angle of elevation from point BB to be 6 degrees. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

Solve a real-world problem using trigonometry: find the distance between two points based on angles of elevation to a lighthouse. The crew measures angles of 15° and 6° at different locations from the lighthouse. Learn how to apply tangent functions and solve for unknown distances. Suitable for high school students in Grades 10-12.

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