Answer
The distance between the two people is increasing at a rate of $~~2.13~mi/h$
Work Step by Step
Let $x$ be the east-west distance between the two people.
Let $y$ be the north-south distance between the two people.
Let $D$ be the distance between the two people.
We can find $x$ after 15 minutes:
$x = (3~mi/h)(0.25~h) - (2~mi/h)~(cos~45^{\circ})(0.25~h)$
$x = 0.396~mi$
We can find $y$ after 15 minutes:
$y = (2~mi/h)~(sin~45^{\circ})(0.25~h)$
$y = 0.354~mi$
We can find $D$ after 15 minutes:
$D^2 = x^2+y^2$
$D = \sqrt{x^2+y^2}$
$D = \sqrt{0.396^2+0.354^2}$
$D = 0.531~mi$
We can find $\frac{dx}{dt}$:
$\frac{dx}{dt} = (3~mi/h) - (2~mi/h)~(cos~45^{\circ})$
$\frac{dx}{dt} = 1.59~mi/h$
We can find $\frac{dy}{dt}$:
$\frac{dy}{dt} = (2~mi/h)~(sin~45^{\circ})$
$\frac{dy}{dt} = 1.41~mi/h$
We can find $\frac{dD}{dt}$:
$D^2 = x^2+y^2$
$2D~\frac{dD}{dt} = 2x~\frac{dx}{dt}+2y~\frac{dy}{dt}$
$\frac{dD}{dt} = \frac{x~\frac{dx}{dt}+y~\frac{dy}{dt}}{D}$
$\frac{dD}{dt} = \frac{(0.396)(1.59)+(0.354)(1.41)}{0.531}$
$\frac{dD}{dt} = 2.13~mi/h$
The distance between the two people is increasing at a rate of $~~2.13~mi/h$
