Jessica and Matthew are running toward the point $P$ along the straight paths that make a fixed angle of $\theta$ (Figure 1).
Suppose that Matthew runs with velocity $v_a~(\mathrm{m/s})$ and Jessica with velocity $v_b~(\mathrm{m/s})$.
Let $f(x,y)$ be the distance from Matthew to Jessica when Matthew is $x$ meters from $P$ and Jessica is $y$ meters from $P$.

$\bullet$ Show that $f(x,y) = \sqrt{x^2 + y^2 - 2xy\cos\theta}$.

$\bullet$ Assume that $\theta = \pi/3$. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica is changing when

$x = 14$, $y = 22$, $v_a = 5~\mathrm{m/s}$, and $v_b = 2~\mathrm{m/s}$.

$\frac{\,df}{\,dt} =$