Final answer:
The distance between the ships is changing at a rate of approximately 6.522 km/h at 4:00 p.m.
Step-by-step explanation:
To find the speed at which the distance between the ships is changing, we can use the concept of relative motion. At 4:00 p.m., 4 hours after noon, ship A would have traveled 120 km (30 km/h x 4 hours) to the east. Ship B, on the other hand, would have traveled 80 km (20 km/h x 4 hours) to the north. Using the Pythagorean theorem, we can calculate the distance between the two ships at 4:00 p.m. as follows:
D = sqrt((170 km + 120 km)^2 + (0 km + 80 km)^2)
= sqrt(57500) km
≈ 239.802 km
To find how fast this distance is changing, we can take the derivative of the distance equation with respect to time:
dD/dt = ((170 km + 120 km)(30 km/h) + (0 km + 80 km)(20 km/h))/(sqrt(57500) km)
Simplifying this expression gives us:
dD/dt ≈ 6.522 km/h
Therefore, the distance between the ships is changing at a rate of approximately 6.522 km/h at 4:00 p.m.