At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 20 knots. How fast (in knots) is the distance between the ships changing at 4 PM?

To find how fast the distance between the ships is changing at 4 PM, we need to find the rate of change of the distance between the ships with respect to time. This can be determined by finding the derivative of the distance between the ships equation with respect to time.

Let's introduce some variables to solve this problem:

Let "x" be the distance (in nautical miles) between ship A and ship B at a given time.
Let "t" be the time elapsed (in hours) since noon.

At noon (t = 0), ship A is 50 nautical miles due west of ship B, so we have x = 50.

Ship A is sailing west at 20 knots, which means its position can be described as A(t) = 50 - 20t, since it starts 50 nautical miles due west and moves 20 nautical miles per hour to the west.

Ship B is sailing north at 20 knots, which means its position can be described as B(t) = 20t, since it starts at the origin and moves 20 nautical miles per hour to the north.

The distance between the ships, x, can be calculated using the distance formula:

x = sqrt((A(t))^2 + (B(t))^2)

x = sqrt((50 - 20t)^2 + (20t)^2)

To find how fast x is changing, we need to take the derivative of x with respect to t:

dx/dt = d/dt(sqrt((50 - 20t)^2 + (20t)^2))

To simplify the calculation, let's assume t = 4 in our equation as we want to find the rate at 4 PM.

dx/dt = d/dt(sqrt((50 - 20 * 4)^2 + (20 * 4)^2))

Now, we can calculate dx/dt to find how fast the distance between the ships is changing at 4 PM.