Question

A boat heading out to sea starts out at Point AA, at a horizontal distance of 1465 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 13degrees
∘
. At some later time, the crew measures the angle of elevation from point BB to be 8degrees
∘
. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

To find the distance from point \( A \) (Point AA) to point \( B \) (Point BB), we can employ some trigonometry.

1. **Calculate the height of the lighthouse** using the angle of elevation from point \( A \) at a distance of 1465 feet.

\[
\tan(\theta) = \frac{\text{height}}{\text{distance}}
\]
Where \( \theta = 13^\circ \) and distance = 1465 feet.

\[
\tan(13^\circ) = \frac{h}{1465}
\]
\[
h = 1465 \cdot \tan(13^\circ)
\]

Now we can calculate the height:
\[
h \approx 1465 \cdot 0.2309 \quad \text{(using a calculator to find } \tan(13^\circ))
\]
\[
h \approx 338.25 \text{ feet}
\]

2. **Using point \( B \)** where the angle of elevation is \( 8^\circ \), we will calculate the distance from point \( B \) to the lighthouse.

Let \( x \) be the distance from point \( B \) to the lighthouse. We can set up the tangent ratio again:
\[
\tan(8^\circ) = \frac{h}{x}
\]
\[
\tan(8^\circ) = \frac{338.25}{x}
\]
\[
x = \frac{338.25}{\tan(8^\circ)}
\]

Calculating \( x \):
\[
x \approx \frac{338.25}{0.1405} \quad \text{(using a calculator for } \tan(8^\circ))
\]
\[
x \approx 2405.34 \text{ feet}
\]

3. **Distance from point \( A \) to point \( B \)**.

We can find the distance between points \( A \) and \( B \) using the total horizontal distance traveled:
\[
d_{AB} = d_A - d_B = 1465 - x
\]
The distance unknown is:
\[
d_{AB} = x_A + x_B = 1465 + x
\]
Applying Pythagorean theorem:
\[
d_{AB}^2 + h^2 = d_B^2
\]
So, to obtain, we have:
\[
d_{AB}^2 = 2405.34^2 - 338.25^2
\]

Calculating:
\[
d_{AB}^2 \approx 5788899.11 - 114100.56
\]
\[
d_{AB} \approx \sqrt{5774798} \approx 2404.3
\]

### Final Result:

Therefore, the distance from point \( A \) to point \( B \) is roughly:

\[
\boxed{2404} \text{ feet} \, \text{(rounded to the nearest foot)}
\]