A man starts walking north at 4 ftys from a point P. Five minutes later a woman starts walking south at 5 ftys from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking?

Let "x" be the current distance travelled by the man from point P after 20 minutes, and "y" be the current distance travelled by the woman after 15 minutes from a point due east of P.

Let "s" be the distance between them after 15 minutes.

dx/dt = rate of change of the man's distance

dy/dt = rate of change of the woman's distance

ds/dt = the rate of change of the distance between the two people.

To calculate the rate of change of the distance between the two of them, we will make use of the product rule:

2 × s × ds/dt = 2(x +y)(dx/dt + dy/dt)

The man walked for a total of 20 minutes and his rate of change of distance = 4ft/s. There are 60 × 20 = 1200 seconds in 20 minutes, so he covered a total of 4ft × 1200sec = 4800ft. This is his current distance after 20 minutes.

The woman walked for 15 minutes and her rate of change of distance = 5ft/s. There are 60 × 15 = 900 seconds in 15 minutes. So she covered a total of 5ft × 900 sec = 4,500ft (This is her current distance after 15 minutes).

Now, we know x, y, dx/dt, and dy/dt but we must determine "s"(current distance between the two people) before we can use the product rule to calculate ds/dt (rate of change of the distance between them).

s^2 = (x + y)^2 + 500(point P to the starting point of the woman)

s^2 = (4800 + 4500)^2 + 500^2

s^2 = 9300^2 + 500^2

s^2 = 86,740,000

s = √86,740,000

s = 9,313.43

Now all that we require to calculate ds/dt is complete.

x = 4800

y = 4500

dx/dt = 4

dy/dt = 5

s = 9,313.43

Fixing these into the product rule:

2 × 9,313 × ds/dt = 2(4800 + 4500) × (4 + 5)

18,626.86 × ds/dt = 2 × 9300 × 9

18,626.86 × ds/dt = 167400

ds/dt = 167400/18626.86

ds/dt = 8.99ft/s

Therefore the rate of change of distance between the man and the woman is 8.99ft/s