an observer stands 300 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 500 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 400 ft from the ground?

Let t be the time since the launch in seconds. Then the height of the rocket will be 500*t and the distance between the observer and the rocket f(t) = sqrt{90000 + 250000t^2}.

The rate of change of the distance equals the derivative f ' (t) = 250000t / sqrt{90000 + 250000t^2}.

The rocket will be 400 ft from the ground at t = 400/500 = 0.8 s.

So, f '(0.8) = 250000*0.8 / sqrt{90000 + 160000} = 200000 / 500 = 400 ft/s.