# Calculus rate of change

**Question description**

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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 120 miles from the point and has a speed of 360 miles per hour. The other is 160 miles from the point and has a speed of 480 miles per hour.

At what rate is the distance between the planes changing?

*s*denote the distance in miles as shown in the following figure. The origin represents the point of convergence.

*s*between the planes is given by the formula,

s =

[img src="http://www.webassign.net/wastatic/wacache1ff8a7b5dc7a7d1f0ed65aaa29c04b1e/watex/img/sqrt2a.gif" > | x^{2} + y^{2} |

s^{2} = x^{2} + y^{2}.

One plane is 120 miles away and moving towards the point of convergence at 360 miles per hour so

*x*= 120and

dx |

dt |

The other plane is 160 miles away and moving towards the point of convergence at 480 miles per hour, so

*y*= 160 and

dy |

dt |

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