At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 22 knots. How fast (in knots) is the distance between the ships changing at 4 PM?

At noon, t=0, A is at (0,0), and B is at (10,0).

A goes due west at 18 knots, and B due north at 22 knots.

The relative velocity vector of B relative to A is Vb-Va=(18,22)

The distance D in nautical miles in terms of time, t hours after noon, between the two ships is expressed by the function:
D(t)=sqrt((10+18t)² + (22t)²)

Thus the rate of change of distance is given by the derivative:
D'(t) = (404t+90)/sqrt(202t^2+90t+25)
and at 4 pm, t=4, and
D'(4) = 28 knots approx.

To find how fast the distance between the ships is changing at 4 PM, we can use the concept of relative velocity.

Let's first figure out the position of each ship at 4 PM:

Ship A: Since it is sailing west at a constant rate of 18 knots, it would have traveled for 4 hours (from noon to 4 PM) and covered a distance of 18 knots/hour * 4 hours = 72 nautical miles. Since it started 10 nautical miles west of ship B, it would be 72 nautical miles + 10 nautical miles = 82 nautical miles west of ship B at 4 PM.

Ship B: It is sailing north at a constant rate of 22 knots, so in 4 hours, it would have traveled 22 knots/hour * 4 hours = 88 nautical miles north.

Now, let's calculate the distance between the ships at 4 PM. We can use the Pythagorean theorem to find the distance:

Distance^2 = (Distance along the west direction)^2 + (Distance along the north direction)^2

Distance^2 = (82 nautical miles)^2 + (88 nautical miles)^2

Distance^2 = 6724 nautical miles^2 + 7744 nautical miles^2

Distance^2 = 14468 nautical miles^2

Distance = sqrt(14468) nautical miles ≈ 120.2 nautical miles

Now, to find how fast the distance between the ships is changing at 4 PM, we can differentiate the distance formula with respect to time:

d(Distance)/dt = (2 * Distance) * (d(Distance)/dt)

Now, substitute the values we know:

d(Distance)/dt = (2 * 120.2 nautical miles) * (d(Distance)/dt)

To find d(Distance)/dt, we need to know the rates at which ship A and ship B are moving horizontally and vertically at 4 PM.

For ship A:
Horizontal speed = -18 knots (westward)
Vertical speed = 0 knots (since it's not moving up or down)

For ship B:
Horizontal speed = 0 knots (since it's not moving left or right)
Vertical speed = 22 knots (northward)

Now, let's substitute the rates of change into the equation:

d(Distance)/dt = (2 * 120.2 nautical miles) * (-18 knots + 0 knots) + (22 knots)

d(Distance)/dt = (2 * 120.2 nautical miles) * (-18 knots + 22 knots)

d(Distance)/dt = (2 * 120.2 nautical miles) * 4 knots

d(Distance)/dt = 960.8 knots

Therefore, the distance between the ships is changing at a rate of approximately 960.8 knots at 4 PM.