Two roads cross at right angles. A car 3 km north of the crossing is driving north, away from it, at 40 km/h; at the same moment a cyclist 4 km east of the crossing is riding west, towards it, at 15 km/h. At what rate is the distance between them increasing at that moment?

Explanation

1. 01

   With the cyclist $x$ km east and the car $y$ km north, $D^{2} = x^{2} + y^{2}$, so $D\frac{dD}{dt} = x\frac{dx}{dt} + y\frac{dy}{dt}$.

2. 02

   Now $x = 4$, $\frac{dx}{dt} = -15$, $y = 3$, $\frac{dy}{dt} = 40$ and $D = 5$: $\frac{dD}{dt} = \frac{-60 + 120}{5} = 12$ km/h.