Radar Path
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What is the speed of the plane?
An airplane is flying on a flight path that will take it directly over a radar tracking station. The plane is flying at an altitude of 6 miles. If the distance between the radar station and the plane (s in the figure) is decreasing at the rate of 400 miles per hour when this distance is 10 miles (when s = 10 miles), what is the speed of the plane?
I’m guessing the diagram should be like this:
. . . . . . . . . . . . . . . . . .|- x -|
· · · · · · · · · · · · · · · · · + – - -|-+ · · · · · · · · · · · · · · ·
^ . . . . . . . . . . . . . . . . | . . /
| . . . . . . . . . . . . . . . . .| . /
6 miles . . . . . . . . . . . .| ./ s
| . . . . . . . . . . . . . . . . .|/
v__________________[#]_________
s = sqrt(x^2 + 6^2)
s^2 = x^2 + 36 => x^2 = s^2 – 36 => x = sqrt(s^2 – 36)
dx/dt = dx/ds * ds/dt
.·. dx/dt = 2s / sqrt(s^2 – 36) * ds/dt
At t0, ds/dt = – 400, s = 10
=> dx/dt = 2(10) / sqrt[(10)^2 - 36] * (- 400)
=> dx/dt = 20 / sqrt(64) * – 400
=> dx/dt = – 8000 / 8
.·. dx/dt = – 1000 mph
.·. Speed = |v| = |dx/dt| = 1000 mph
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