To solve this problem, we can use trigonometry, specifically the tangent of angles, to find the height of the lighthouse and then apply the height to find the distance between points $A$ and $B$.

Step 1: Set up known values and define variables

1. Let $h$ be the height of the lighthouse.
2. Let $x$ be the distance between Point $A$ and Point $B$.
3. The distance from the lighthouse to Point $A$ is 1465 feet.
4. The angle of elevation from Point $A$ to the beacon is 13°.
5. The angle of elevation from Point $B$ to the beacon is 8°.

Step 2: Use the tangent function to find $h$

From Point $A$: $\tan(13^\circ) = \frac{h}{1465}$ Solving for $h$: $h = 1465 \cdot \tan(13^\circ)$

Step 3: Calculate $h$

Using $\tan(13^\circ) \approx 0.2309$: $h \approx 1465 \cdot 0.2309 \approx 338.06 \text{ feet}$

Step 4: Use $h$ to find the distance $x$

From Point $B$, we set up a similar equation using the angle of elevation: $\tan(8^\circ) = \frac{h}{1465 + x}$ Substitute $h \approx 338.06$: $\tan(8^\circ) = \frac{338.06}{1465 + x}$ Using $\tan(8^\circ) \approx 0.1405$, we solve for $x$: $0.1405 = \frac{338.06}{1465 + x}$ Rearrange and solve for $x$: $1465 + x = \frac{338.06}{0.1405}$ $1465 + x \approx 2406.97$ $x \approx 2406.97 - 1465$ $x \approx 942 \text{ feet}$

Answer

The distance from Point $A$ to Point $B$ is approximately 942 feet.

Would you like a deeper explanation of any step?

---

Here are some related questions to explore further:

1. How would the answer change if the angle at Point $A$ were 15° instead of 13°?
2. What would be the height of the lighthouse if the boat's angle of elevation at both points was the same?
3. How would using a different trigonometric function, such as cosine, affect our approach to this problem?
4. What if the initial distance from the shore was given in meters? How would you convert it to feet?
5. Could we solve this problem using the sine function instead? Why or why not?

Tip: For navigation problems like this, trigonometric functions are very effective when dealing with right-angled scenarios and elevations.