Final answer:
The ship's distance from port is 3.24 nautical miles, and its direction is 82° west of north.
Step-by-step explanation:
To determine the ship's distance and direction from port to its current location, we can use vector addition. We start by representing the initial heading of N 60° W as a vector. This vector can be broken down into two components: one along the north direction and one along the west direction. The north component can be found using the sine function: sin(60°) = north component / 12 nautical miles.
Solving for the north component gives us 10.4 nautical miles. Similarly, the west component can be found using the cosine function: cos(60°) = west component / 12 nautical miles.
Solving for the west component gives us 6 nautical miles.
Next, we represent the change in course to a bearing of N25° E as another vector.
Again, this vector can be broken down into two components: one along the north direction and one along the east direction. The north component can be found using the cosine function: cos(25°) = north component / 15 nautical miles. Solving for the north component gives us 13.6 nautical miles.
Similarly, the east component can be found using the sine function: sin(25°) = east component / 15 nautical miles. Solving for the east component gives us 6.5 nautical miles.
To find the ship's distance and direction from port, we add the respective components together.
The north component is 10.4 nautical miles - 13.6 nautical miles = -3.2 nautical miles. The east component is 6.5 nautical miles - 6 nautical miles = 0.5 nautical miles.
To find the total distance, we use the Pythagorean theorem: distance = sqrt((-3.2)^2 + (0.5)^2) = 3.24 nautical miles.
To find the direction, we use the inverse tangent function: angle = atan((-3.2/0.5))
= -82°.
Therefore, the ship is 3.24 nautical miles away from port and heading in a direction of 82° west of north.