Aplane flying horizontally at an altitude of 1 mile and a speed of of 500mih passes directly over a radar station. find the rate at which the distance from the plane to the station is increasing when it is 2mi away from the station.

447.2 mph

Explanation:

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

Therefore:

√5 dd/dt = (2) (500)

dd/dt = 1000/√5

dd/dt = 200√5

dd/dt ≈ 447.2 mph

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