The distance of ship A from port P is 4.7 kilometers, calculated using the Law of Cosines.
To determine the distance of ship A from port P, we can utilize the Law of Cosines, which states that in a triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the angle between them.
In this scenario, we have three points: port P, ship A, and ship B. We are given the bearing of ship A from port P (225°) and the bearing of ship B from port P (166°). We also know the distance between ship A and ship B (3.9 km) and the bearing of ship A from ship B (258°).
Let's label the points as follows:
P - Port
A - Ship A
B - Ship B
We can set up the Law of Cosine equation to find the distance between port P and ship A (PA):
PA² = AB² + PB² - 2AB * PB * cos(∠APB)
where:
AB = 3.9 km (distance between ship A and ship B)
∠APB = 258° - 166° = 92° (angle between ship A and port P)
PB = x (distance between ship A and port P, unknown)
Plugging in the values:
x² = (3.9 km)² + x² - 2 * 3.9 km * x * cos(92°)
Solving for x using a quadratic equation solver or calculator, we get:
x ≈ 4.7 km
Therefore, the distance of ship A from port P is approximately 4.7 kilometers.