What rate is the distance changing and how much times does the controller have to take to get on a different path?

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 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? ______mph

(b)How much time does the controller have to get one of the airplanes on a different flight path? _____mph

Answer 1

(X^2 + y^2)^(1/2) = D(x,y) by pythagorean theorem

where x and y are functions of time

D' = (1/2) ( x^2 + y^2) ^(-1/2) (2x dX/dt + 2y dY/dt)

= (x dX/dt + y * dY/dt) / ( x^2 + y^2)^(1/2) <--- 2 factors out and cancels 1/2

x= (150*300 + 200*400)/ sqrt( 150^2 + 200^2) = 125000 / 250 = 500

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200 - 400t = 150 - 300t

200 - 150 = -300t +400 t

50 = 100t

t = 1/2

They will collide in 1/2 hour = 30 minutes