Question

An airplane is flying 600 mph on a horizontal path that will take it directly over an observer. The airplane remains a constant altitude of 7 miles. What is the rate of change of the distance between the observer and the airplane when X = 5 miles?

Answer 1

Change of the distance between the observer and the airplane when X = 5 miles is 348.84 mph

Step-by-step explanation:

We have by Pythagoras theorem

       Distance between the observer and the airplane² = Horizontal distance² + Altitude²

        d² = h² + a²

Here we have

          h = X = 5 miles

          a = 7 miles

          d² = 5² + 7²  = 74

          d =8.60 miles

          $\frac{dh}{dt}=600mph\\\\\frac{da}{dt}=0$

Differentiating d² = h² + a² with respect to time

            $2d\frac{dd}{dt}=2h\frac{dh}{dt}+2a\frac{da}{dt}\\\\8.60\frac{dd}{dt}=5\times 600+7\times 0\\\\\frac{dd}{dt}=348.84mph$

Change of the distance between the observer and the airplane when X = 5 miles is 348.84 mph