Question

A boat heading out to sea starts out at Point AA, at a horizontal distance of 11151115 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light to be 7, degrees7
∘
. At some later time, the crew measures the angle of elevation from point BB to be 2, degrees2
∘
. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

Answer 1

To solve this problem, we can use trigonometry. We will analyze both scenarios using right triangles formed by the heights of the lighthouse and the horizontal distances from points A and B.

1. **From point A:**
- Distance from point A to the lighthouse (horizontal distance) = 11151115 feet.
- Angle of elevation = \( 7^\circ \).

We can find the height of the lighthouse (h) using the tangent function:
\[
\tan(7^\circ) = \frac{h}{11151115}
\]
Rearranging gives us:
\[
h = 11151115 \cdot \tan(7^\circ)
\]

Now we calculate \( h \):
\[
\tan(7^\circ) \approx 0.122784
\]
Hence,
\[
h \approx 11151115 \cdot 0.122784 \approx 1361623.013
\]

2. **From point B:**
- Let \( d \) be the horizontal distance from point B to the lighthouse.
- Angle of elevation from point B = \( 2^\circ \).

Again, using the tangent function, we have:
\[
\tan(2^\circ) = \frac{h}{d}
\]
Rearranging gives:
\[
d = \frac{h}{\tan(2^\circ)}
\]

Now let's calculate \( d \):
\[
\tan(2^\circ) \approx 0.034921
\]
Thus,
\[
d \approx \frac{1361623.013}{0.034921} \approx 38975816.97 \text{ feet}
\]

3. **Finding distance from A to B:**
The distance from Point A to Point B is the difference between the two horizontal distances:
\[
\text{Distance from A to B} = d - 11151115
\]
Therefore,
\[
\text{Distance from A to B} \approx 38975816.97 - 11151115 \approx 27824701.97 \text{ feet}
\]

4. **Rounding the answer:**
Finally, rounding to the nearest tenth:
\[
\text{Distance from A to B} \approx 27824702.0 \text{ feet}
\]

So, the distance from Point A to Point B is approximately **27824702.0 feet**.