Hello, FalconPUNCH!

Make a sketch!

1) A man starts walking north at 4ft/s from a point P.

Five minutes later a woman starts walking south at 5ft/s from a point 500ft due east of P.

At what rate are the people moving apart 15 min after the woman starts walking? Code:

M *
| *
4t | *
| *
A * *
| *
| *
1200 | *
| *
| *
P * - - - - - - - - - * - - - * Q
: * |
5t : * | 5t
: * |
R * - - - - - - - - - - - - - * W
500

The man starts at P and has a 5-minute headstart.

In 5 minutes (300 sec), he walks 1200 feet to point A.

In the next $\displaystyle t$ seconds, he walks $\displaystyle 4t$ ft to point M.

The woman starts at Q and walks south at 5 ft/s.

In the next $\displaystyle t$ seconds, she walks $\displaystyle 5t$ ft to point W.

Let $\displaystyle x \,=\,MW$

In right triangle MRW, we have: .$\displaystyle MW^2 \:=\:MR^2 + RW^2$

. . That is: .$\displaystyle x^2\:=\4t+1200 + 5t)^2 + 500^2 \:=\9t+1200)^2 + 500^2$

Differentiate with respect to time: .$\displaystyle 2x\left(\frac{dx}{dt}\right) \:=\:2(9t + 1200) $

. . Hence: .$\displaystyle \frac{dx}{dt} \;=\;\frac{9t + 1200}{x}$

In 15 minutes, $\displaystyle t = 900$

Find $\displaystyle x$ at that time, and you're done!