This is a classic trigonometry problem involving right triangles, angles of elevation, and distances. Let's break it down and solve it step-by-step.

Problem Summary:

• The boat starts at point $A$, 1032 feet horizontally from the lighthouse.
• At point $A$, the angle of elevation to the lighthouse is $15^\circ$.
• The boat moves farther out to point $B$, where the angle of elevation is $6^\circ$.
• We need to find the distance from point $A$ to point $B$.

Step 1: Establish Variables and Use of Trigonometry

Let's define:

• $h$ as the height of the lighthouse.
• $d$ as the distance from point $A$ to point $B$.

From point $A$, we can use the tangent function since we know the angle of elevation and the horizontal distance: $\tan(15^\circ) = \frac{h}{1032}$ From point $B$, we know the angle of elevation is $6^\circ$, and the total horizontal distance from the lighthouse is $1032 + d$: $\tan(6^\circ) = \frac{h}{1032 + d}$

Step 2: Solve for the Height $h$

Using the equation from point $A$: $h = 1032 \times \tan(15^\circ)$ Now, I'll calculate the height $h$.

Step 3: Substitute $h$ into the Equation for Point $B$

Once we have the height, we can substitute it into the equation for point $B$ to find $d$: $\tan(6^\circ) = \frac{h}{1032 + d}$ I'll now solve for $d$.

Let me calculate these values for you.The height of the lighthouse $h$ is approximately 276.5 feet.

The distance from point $A$ to point $B$ is approximately 1598.9 feet.

Would you like more details or clarifications on the steps?

Here are five related questions to expand on this topic:

1. How do we use trigonometry to solve right triangle problems in different contexts?
2. What are the differences between angle of elevation and angle of depression?
3. How can we apply the tangent function to solve real-life navigation problems?
4. What are the limitations of using trigonometric functions in distance measurement?
5. How do rounding errors affect trigonometric calculations?

Tip: When working with angles, always ensure your calculator is set to the correct mode (degrees or radians) to avoid errors.