Question

An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other. One airplane is 70 miles from the point and has a speed of 140 miles per hour. The other is 240 miles from the point and has a speed of 480 miles per hour.
(a) At what rate is the distance between the planes changing?
(b) How much time does the controller have to get one of the airplanes on a different flight path?

Answer 1

So.... notice the picture below

we know when y = 140 and x = 240, r = 250

so let's use the pythagorean theorem

$\bf r^2=x^2+y^2\implies 2r\cfrac{dr}{dt}=2x\cfrac{dx}{dt}+2y\cfrac{dy}{dt}\implies \cfrac{dr}{dt}=\cfrac{x\frac{dx}{dt}+y\frac{dy}{dt}}{r} \\\\\\ \cfrac{dr}{dt}=\cfrac{240\cdot 480+70\cdot 140}{250}\implies \cfrac{dr}{dt}=500$

now... how much time does the controller have? well, say, plane "y" is going at dy/dt of 140mph, and is 70 miles away, 140 is 2*70, so it can cover those 70 miles in 1/2 hr, and when it does, it'll collide with the other,

so, the controller has that much, less than 1/2 hr