# What is the rate of change?

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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.

Part (a)

At what rate is the distance between the planes changing?

Step 1 of 5

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

The distance s between the planes is given by the formula,

s =

 [img src=”http://www.webassign.net/wastatic/wacache1ff8a7b5dc7a7d1f0ed65aaa29c04b1e/watex/img/sqrt2a.gif” > x2 + y2

.

Thus,

s2 = x2 + y2.

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

= −   .