Answer
The people are moving apart at a rate of $8.987~ft/s$
Work Step by Step
Let $x$ be the east-west distance between the two people.
Then $x = 500~ft$
Let $y$ be the north-south distance between the two people. We can find $y$ fifteen minutes after the woman starts walking:
$y = (4~ft/s)(20 \cdot 60~s)+ (5~ft/s)(15\cdot 60~s)$
$y = 9300~ft$
Let $z$ be the distance between the two people. We can find $z$ fifteen minutes after the woman starts walking:
$z^2 = x^2+y^2$
$z = \sqrt{x^2+y^2}$
$z = \sqrt{(500~ft)^2+(9300~ft)^2}$
$z = 9313.43~ft$
We can differentiate both sides of the equation with respect to $t$:
$z^2 = x^2+y^2$
$2z~\frac{dz}{dt} = 2x~\frac{dx}{dt} + 2y~\frac{dy}{dt}$
$\frac{dz}{dt} = \frac{1}{z}~(x~\frac{dx}{dt} + y~\frac{dy}{dt})$
$\frac{dz}{dt} = \frac{1}{9313.43}~[(500)(0) + (9300)(4+5)]$
$\frac{dz}{dt} = 8.987~ft/s$
The people are moving apart at a rate of $8.987~ft/s$.