Question

A boat heading out to sea starts out at Point AA, at a horizontal distance of 1433 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.
Answer
Attempt 1 out of 2

Answer 1

To find the distance between points AA and BB, we can start by determining the height of the lighthouse using the information given for point AA.

1. **Finding the height of the lighthouse (h) using Point AA**:

From point AA, we have:
- Horizontal distance (d) = 1433 feet
- Angle of elevation (θ = 15°)

Using the tangent function:
\[
\tan(15°) = \frac{h}{1433}
\]
Thus,
\[
h = 1433 \times \tan(15°)
\]
Now, calculating \( \tan(15°) \):
\[
\text{tan(15°)} \approx 0.2679
\]
Now substituting:
\[
h = 1433 \times 0.2679 \approx 384.1 \text{ feet}
\]

2. **Finding the distance from Point BB**:

Now, from point BB, the angle of elevation is 6°. We can again use the tangent function, with the height \( h \) we found:
\[
\tan(6°) = \frac{h}{d_{BB}}
\]
Where \( d_{BB} \) is the horizontal distance from point BB to the lighthouse.

Using the known height:
\[
\tan(6°) \approx 0.1051
\]
So,
\[
0.1051 = \frac{384.1}{d_{BB}}
\]
Now rearranging to find \( d_{BB} \):
\[
d_{BB} = \frac{384.1}{0.1051} \approx 3646.0 \text{ feet}
\]

3. **Calculating the distance between Point AA and Point BB**:

The distance between Point AA and Point BB is simply the difference in horizontal distances to the lighthouse:
\[
d_{AA to lighthouse} - d_{BB to lighthouse} = 3646.0 - 1433 = 2213.0 \text{ feet}
\]

Finally, rounding to the nearest tenth, the distance from point AA to point BB is approximately:
\[
\boxed{2213.0} \text{ feet}
\]