Answer the following question.

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 420 miles per hour. The other is 240 miles from the point and has a speed of 1440 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

This is a related rates question, where the distance between the planes is a function of the leg lengths of the right triangle with vertices at the two planes and the point of convergence. Since we know the rates at which the leg lengths are changing, we can determine the rate of change of the distance between the two planes (i.e. the hypotenuse):

x: slower plane's leg (mi) ; y: faster plane's leg (mi) ; z: distance between planes (mi)

x² + y² = z² Differentiating with respect to time gives ...

2x dx/dt + 2y dy/dt = 2z dz/dt and dx/dt = - 420 , dy/dt = - 1,440 , x = 70 , y = 240 , z = 250

dz/dt = [(70)(-420) + (240)(-1440)] / 250 = - 1,500 mph

Looking at either leg or the hypotenuse, we can see that the speed is 6 times the distance, so the air traffic controller has 1/6 hr (10 minutes) to reroute one of the planes, though probably the sooner the better.