A plane flying horizontally at an altitude of 2 mi and a speed of 570 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 5 mi away from the station. (Round your answer to the nearest whole number.)

Step-by-step explanation:

Rate of Change

The instantaneous rate of change is computed as the derivative of a function with respect to an independent variable, usually time.

We know a plane is flying horizontally at an altitude h=2 miles when it passes directly over a radar station. Let's call x the horizontal distance the plane travels.

Since the change of the distance x over time is the speed, then x'=570 mi/h. The distance x, the height and the distance between the plane and the radar station (y) form a right triangle. By the Pythagora's theorem

We need to compute the rate at which y is changing when y=5 mi.

Differentiating with respect to time:

Please note that the altitude is constant, thus h'=0 and:

Solving for y'

We need to compute x when y=5 and h=2

Thus we have

Fiinally

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).