Question

A boat is heading towards a lighthouse, whose beacon-light is 131131 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 15, degrees15 ∘ , before they draw closer. They measure the angle of elevation a second time from point BB to be 25, degrees25 ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

To solve this problem, we can use the tangent function from trigonometry, which relates angles and distances in a right triangle.

### Given information:
- Height of the lighthouse (beacon light) = 131131 feet
- Angle of elevation from point A (α) = 15 degrees
- Angle of elevation from point B (β) = 25 degrees

### Step 1: Determine the distances from points A and B to the lighthouse.
Let's denote:
- \( d_A \) = distance from point A to the lighthouse
- \( d_B \) = distance from point B to the lighthouse

Using the tangent function:

\[
\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d_A}
\]
\[
\tan(15^\circ) = \frac{131131}{d_A}
\]

Solving for \( d_A \):

\[
d_A = \frac{131131}{\tan(15^\circ)}
\]

Calculating \( d_A \):

\[
d_A \approx \frac{131131}{0.2679} \approx 489000.049
\]
So,

\[
d_A \approx 489000.05 \text{ feet}
\]

Now for point B, similarly:

\[
\tan(\beta) = \frac{h}{d_B}
\]
\[
\tan(25^\circ) = \frac{131131}{d_B}
\]

Solving for \( d_B \):

\[
d_B = \frac{131131}{\tan(25^\circ)}
\]

Calculating \( d_B \):

\[
d_B \approx \frac{131131}{0.4663} \approx 281000.169
\]
So,

\[
d_B \approx 281000.17 \text{ feet}
\]

### Step 2: Find the distance from point A to point B.
The distance from point A to point B (\( d_{AB} \)) can be found using:

\[
d_{AB} = d_A - d_B
\]

Calculating \( d_{AB} \):

\[
d_{AB} = 489000.05 - 281000.17 \approx 208999.88
\]

### Final Result:
Rounding this to the nearest foot, we find:

\[
\boxed{209000} \text{ feet}
\]

So, the distance from point A to point B is approximately 209,000 feet.