An air traffic controller spots two airplanes at the same altitude converging to a point as they fly at right angles to each other.

Question

One airplane is 150 miles from the point and has a speed of 300 miles per hour. The other is 200 miles from the point and has a speed of 400 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

Visualize the two planes on the XY axis.

You are at the origin.

Plane X: The first plane is due west of you, or at x = -150

traveling towards you at 300 MPH

Plane Y: The second plane is due south of you, or at y = -200

traveling towards you at 400 MPH.

D = Distance between the two planes.

D = sqrt ( X^2 + Y^2 )

d/dt [ D ] = d/dt [ sqrt ( X^2 + Y^2 ) ]

dD/dt = (1/2)[ X^2 + y^2 ]^(-1/2) * [ 2XX' + 2YY' ]

D' = (1/2)[ (-150)^2 + (-200)^2 ]^(-1/2) * [ 2(-150)(300) + 2(-200)(400) ]

D' = [ (-150)^2 + (-200)^2 ]^(-1/2) * [ (-150)(300) + (-200)(400) ]

D' = [ 62500 ]^(-1/2) * [ -125000 ]

D' = (1/250) * [ -125000 ]

D' = - 500

Part (a)

The distance is decreasing at a rate of 500 miles/hr

Part (b)

Recall, D = Rt ==> t = D/R

The X-Plane reaches the origin at t = 150/300 = 1/2 hour.

The Y-Plane reaches the origin at t = 200/400 = 1/2 hour

So, in 1/2 hour, they both will reach the same spot (you, at the origin).

So, the controller has less than 1/2 hour to divert the plane(s).