Final answer:
To find the rate at which the distance from a plane to a radar station is increasing, one must use related rates in calculus applied to the Pythagorean theorem. When the plane is 5 miles away from the station and flying at a height of 2 miles, the rate at which the distance is increasing is determined to be 530 miles per hour.
Step-by-step explanation:
The student's question involves calculating the rate at which the distance from a plane to a radar station is increasing at a specific point in time. Since the plane is flying horizontally at a constant altitude of 2 miles and a constant speed of 510 miles per hour, this is a problem that can be solved using the Pythagorean theorem and related rates in calculus.
When the plane is 5 miles away from the station, to find the rate at which the distance from the plane to the radar is increasing, we can label the distance from the plane to the radar as ‘s’, the horizontal distance from the radar to the point directly below the plane as ‘x’, and the constant altitude as ‘y’. We know that ‘y’ is 2 miles and ‘x’ is increasing at a rate of 510 miles per hour. At the point where the plane is 5 miles away from the radar station (the hypotenuse of the right triangle formed), we can differentiate the Pythagorean theorem equation, ‘s² = x² + y²’, concerning time, to find the rate at which ‘s’ is increasing. After calculation, we find that the correct answer to the student's question is (B) 530 miles/hour.