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# Compute the velocity of an airplane given the rate of change of its distance to a point on the ground

An airplane is 8 miles above the ground flying at a constant velocity, at a constant altitude. (Assume the earth is flat.) There is a point on the ground directly under the airplane’s flight path. The distance between and the airplane is decreasing at a rate of 4 miles per minute when the distance is 10 miles. Find the velocity of the airplane.

Let be the distance from the point on the ground directly beneath the plane to the point , and let be the distance from the plane to . The following diagram illustrates the situation:

We are trying to compute , the velocity of the airplane. We are given that the distance from the airplane to the point is changing at a rate of 4 miles per minute when . Thus, adjusting units to mph, we have

Furthermore, we can compute in terms of (by the Pythagorean identity) and then differentiate:

So, at miles we have ; thus,

This was computed for miles, but since we are given that the airplane is flying at a constant velocity, then this velocity is valid for all .