To solve this problem, we can use the concept of related rates. We'll start by finding the position of each ship at 6 PM, and then calculate the rate at which the distance between them is changing.
Step 1: Calculate the distance traveled by each ship by 6 PM.
Assuming that the coordinates for ship A's position are (0, 0), we can calculate the positions of ship A and ship B at 6 PM using their respective speeds and the time difference.
Ship A: Since ship A is sailing west at a speed of 18 knots for 6 hours, it would have traveled a distance of 18 × 6 = 108 nautical miles due west.
Ship B: Similarly, since ship B is sailing north at a speed of 22 knots for 6 hours, it would have traveled a distance of 22 × 6 = 132 nautical miles due north.
Step 2: Calculate the coordinates of ship B at 6 PM.
Since ship A starts at the origin (0, 0) and ship B is 10 nautical miles due west, ship B's final x-coordinate at 6 PM would be -10, while its y-coordinate would be 0 + 132 = 132.
Step 3: Calculate the distance between the ships at 6 PM.
Using the distance formula, we can find the distance between the two ships at 6 PM.
Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Distance = sqrt[(-10 - 0)^2 + (132 - 0)^2]
Distance = sqrt[100 + 17424]
Distance ≈ sqrt(17524)
Distance ≈ 132.369 nautical miles
Step 4: Calculate the rate at which the distance between the ships is changing at 6 PM.
To find the rate at which the distance between the ships is changing, we can differentiate the distance equation with respect to time and evaluate it at 6 PM.
The rate of change of distance between the ships = d(Distance)/dt
Using the chain rule of differentiation,
d(Distance)/dt = [d(Distance)/dx] * [dx/dt] + [d(Distance)/dy] * [dy/dt]
Since dx/dt represents the westward velocity of ship A (18 knots) and dy/dt represents the northward velocity of ship B (22 knots), we can substitute these values.
d(Distance)/dt = [d(Distance)/dx] * 18 + [d(Distance)/dy] * 22
To find the rates of change of distance with respect to x and y, we can differentiate the distance equation:
d(Distance)/dx = (x2 - x1) / Distance
d(Distance)/dy = (y2 - y1) / Distance
Substituting the values obtained earlier:
d(Distance)/dx = (-10 - 0) / 132.369 ≈ -0.075 nautical miles
d(Distance)/dy = (132 - 0) / 132.369 ≈ 1 nautical mile
Substituting the derived values:
d(Distance)/dt = (-0.075) * 18 + (1) * 22
d(Distance)/dt ≈ -1.35 + 22
d(Distance)/dt ≈ 20.65 knots
Therefore, the distance between the ships is changing at a rate of approximately 20.65 knots at 6 PM.