To find the distance between pier A and pier B, we can use trigonometry, specifically the Law of Sines.
Given:
- The boat distance from the shore: 200 feet.
- The angle from the boat to pier A: \( 35^\circ \).
- The angle from the boat to pier B: \( 75^\circ \).
1. Convert angles to radians:
- Angle to pier A in radians: \( 0.6108652381980153 \) radians.
- Angle to pier B in radians: \( 1.3089969389957472 \) radians.
2. Find the angle between pier A and pier B along the shore:
- Angle between the piers: \( 75^\circ - 35^\circ = 40^\circ \).
- Converted to radians: \( 0.6981317007977318 \) radians.
3. Apply the Law of Sines:
We have three points: the boat (P), pier A (A), and pier B (B). The angle at point P between the line of sights to piers A and B is the angle we found above (40° or \(0.6981317007977318\) radians).
Using the Law of Sines:
[tex]\[
\frac{\sin(\text{angle A})}{d_{PA}} = \frac{\sin(\text{angle B})}{d_{PB}} = \frac{\sin(\text{angle between A and B})}{\text{boat distance}}
\][/tex]
We need to find \(d_{AB}\), the distance between pier A and pier B:
[tex]\[
d_{AB} = \text{boat distance} \times \frac{\sin(\text{angle between piers})}{\sin(180^\circ - \text{angle B})}
\][/tex]
Since \( \sin(180^\circ - \text{angle B}) = \sin(\text{angle B}) \):
[tex]\[
d_{AB} = 200 \times \frac{\sin(40^\circ)}{\sin(75^\circ)}
\][/tex]
Simplifying further:
[tex]\[
d_{AB} \approx 133 \text{ feet}
\][/tex]
Thus, the distance between pier A and pier B, rounded to the nearest foot, is:
[tex]\[
\boxed{133}
\][/tex]
Pier A is 133 feet from pier B.
