Final answer:
To find how fast the distance between Ship A and Ship B is changing at 4:00pm, use the Pythagorean theorem and differentiate. With Ship A moving south at 40km/hr and Ship B moving north at 20km/hr for 4 hours, the distance is changing at a rate of 60 km/hr at that time.
Step-by-step explanation:
To determine how fast the distance between the ships is changing at 4:00pm, we should use the Pythagorean theorem to relate the distances.
At noon, Ship A is 100km west of Ship B. After 4 hours, Ship A would have moved 4 hours * 40 km/hr = 160 km south, and Ship B would have moved 4 hours * 20 km/hr = 80 km north. Now we have a right triangle where Ship A has moved 160 km from its original position, Ship B has moved 80 km from its original position, and the initial distance between the ships was 100 km due west.
The distance between the ships at 4:00 pm can be computed using the Pythagorean theorem: distance = √(100^2 + (160+80)^2) = √(100^2 + 240^2). The change in distance over time is the derivative of this distance with respect to time, which involves the rate of change of the east-west distance (which is 0 since neither ship is moving east or west) and the rate of change of the north-south distance (which is the sum of the speeds of the two ships in the north-south direction).
Therefore, the rate at which the distance between the ships changing at 4:00pm is the sum of the southward speed of ship A and the northward speed of ship B, which is 40 km/hr + 20 km/hr = 60 km/hr.