At noon, ship A is 150 km west of ship B. Ship A is sailing east at 20 km/h and ship B is sailing north at 25 km/h. After two hours, they are approximately 121 km apart. How fast is the distance between the ships changing after two hours?
We can use Pythagoras:
$\displaystyle D^{2}=x^{2}+y^{2}$
After two hours, A has traveled 40 km and B has traveled 50 km.
Therefore, At two hours the distance between them is:
$\displaystyle \sqrt{(150-40)^{2}+50^{2}}=10\sqrt{146}\approx{120.83} \;\ km$
differentiate:
$\displaystyle D\frac{dD}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}$
$\displaystyle 10\sqrt{146}\frac{dD}{dt}=110(20)+50(25)$
$\displaystyle \frac{dD}{dt}=\frac{345}{\sqrt{146}}\approx{28.55} \;\ km/hr$