The vertical distance from the plane to the station is 1 mile.
The horizontal distance from the plane to the station is x miles.
The distance between the plane and station is therefore:
d² = 1² + x²
Taking derivative with respect to time:
2d dd/dt = 0 + 2x dx/dt
d dd/dt = x dx/dt
We know that x = 2 miles and dx/dt = 500 mi/hr. We need to find d when x=2.
d² = 1² + 2²
d² = 5
d = √5
√5 dd/dt = (2) (500)
dd/dt = 1000/√5
dd/dt = 200√5
dd/dt ≈ 447.2 mph