A man starts walking north at 3 ft/s from a point P. Five minutes later a woman starts walking south at 4 ft/s from a point 500 ft due east of P. At what rate (in ft/s) are the people moving apart 15 minutes after the woman starts walking? (Round your answer to two decimal places.)

This problem involves the Pythagorean theorem and rates of change. Here are the steps to solve it:

1. First, we need to determine how far each person has walked. The man has been walking for 20 minutes (15 minutes after the woman started plus the 5 minutes he started earlier). Since he walks at 3 ft/s, he has walked 3 ft/s * 60 s/min * 20 min = 3600 feet.

2. The woman has been walking for 15 minutes at 4 ft/s. So, she has walked 4 ft/s * 60 s/min * 15 min = 3600 feet.

3. Now, we need to find the distance between them. We know that the woman started 500 feet east of the man, and they are walking in opposite directions. So, we can use the Pythagorean theorem to find the distance between them. The distance is sqrt((3600 ft)^2 + (500 ft)^2) = approximately 3627.67 feet.

4. The rate at which they are moving apart is the derivative of this distance with respect to time. We can use the chain rule to find this. The derivative of sqrt(x^2 + y^2) with respect to x is x/sqrt(x^2 + y^2), and the derivative of sqrt(x^2 + y^2) with respect to y is y/sqrt(x^2 + y^2).

5. So, the rate at which they are moving apart is (3600 ft/s)/sqrt((3600 ft)^2 + (500 ft)^2) + (500 ft/s)/sqrt((3600 ft)^2 + (500 ft)^2) = approximately 4.99 ft/s.

So, 15 minutes after the woman starts walking, the people are moving apart at a rate of approximately 4.99 ft/s.