Well, isn't that a "ship-stick" situation? Let's "sail" into the problem.
To find the rate at which the distance between the ships is changing, we need to consider their respective velocities. Ship A is sailing south at 30 km/h, while Ship B is sailing north at 15 km/h.
Since they are moving in opposite directions, we can simply add their velocities to find the relative speed between the two ships. The relative speed is 30 km/h + 15 km/h = 45 km/h.
Given that the ships were initially 100 km apart, we can now calculate how fast the distance between them is changing. We're looking for the rate of change at 4:00pm, so let's put on our "time-traveler" hats and calculate.
From noon until 4:00pm, there are 4 hours. Multiplying the relative speed by the duration gives us the change in distance between the ships: 45 km/h * 4 h = 180 km.
Therefore, the distance between the ships is changing at a rate of 180 km over the course of 4 hours, or 45 km/h.