At noon, ship A is 100 km west of ship B. Ship A sails east at 20 km/h and ship B sails north at 15 km/h. How fast is the distance between them changing at 2 PM?

Core Concept

Let $z$ be the distance. From $z^2=(100-20t)^2+(15t)^2$, at $t=2$ hours, $z=30\sqrt{5}$ and $z\dfrac{dz}{dt}=60(-20)+30(15)=-750$. Thus $\dfrac{dz}{dt}=-\dfrac{750}{30\sqrt{5}}=-5\sqrt{5}$, so the distance decreases at $5\sqrt{5}$ km/h.