"a plane flying horizontally at an altitude of 1 mi and a speed of 520 mi/h passes directly over a radar station. find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (round your answer to the nearest whole number.)"

I'm guessing "distance from the plane to the station" refers to the Euclidean distance, or how far away the plane is in space from the station, and not horizontal distance, or how far it would be from the station if it were traveling at the same altitude as the station.

Let be the distance from the plane to the station. The plane forms a right triangle with the station with one leg of length 1 mi. The second leg's length can be determined from by the plane's current distance via the Pythagorean theorem:

So when the plane's distance is , we have .

We know that changes at the same rate as the plane's speed, i.e. . Differentiating the equation above with respect to time , we have

At the point when , we found that , and we're given , so we find

astronomers is your final answer! : )