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AIR TRAFFIC CONTROL: and airplane is flying at an altitude of 5 miles and passes directly over a radar antenna?
When a plane is 10 miles away (s=10), the radar detects that the distance s is changing at a rate of 240mi/hr What is the speed of the plane?
Let s denote the distance between the radar and the airplane and x denote the horizontal distance between the radar and the plane.
Since the airplane is flying at a height of 5 miles, by the Pythagorean Theorem:
x^2 + 5^2 = s^2 ==> s^2 = x^2 + 25.
By differentiating implicitly with respect to time:
(d/dt)(s^2) = (d/dt)(x^2 + 25) ==> 2s(ds/dt) = 2x(dx/dt).
Solving this for dx/dt gives:
dx/dt = (s/x)(ds/dt).
With s = 10, we see that:
10^2 = x^2 + 25 ==> x^2 = 75 and x = 5√3.
Therefore:
dx/dt = (s/x)(ds/dt) = [10/(5√3)](240) ≈ 277 mph.
I hope this helps!
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Standing next to a radar antenna
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