A police helicopter is hovering at a constant altitude of 0.5 mile above a strai

Question

A police helicopter is hovering at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and this distance is decreasing at 66 mph. Find the speed of the car. A police helicopter is hovering at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and this distance is decreasing at 66 mph. Find the speed of the car. A police helicopter is hovering at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and this distance is decreasing at 66 mph. Find the speed of the car.

Explanation / Answer

Ans) The distance (z) between the helicopter and the car forms the hypotenuse of a right triangle. The altitude is constant (0.5 mi) and the other leg (x, along the ground) is decreasing due to the moving helicopter and the moving car.

z^2 = x^2 + (0.5 mi)^2

Differentiate implicitly with respect to time:
2z(dz/dt) = 2x(dx/dt)
z(dz/dt) = x(dx/dt)

For the moment in question, z = 1 mi, dz/dt = -66 mi/hr, and x would be:
1^2 = x^2 + (0.5)^2
x = sqrt(3)/2 mi

dx/dt represents how the "ground" distance is changing. So we solve for it.
(1 mi)(-66 mi/hr) = (sqrt(3)/2)(dx/dt)
dx/dt = 76.21 mi/hr

Subtract the speed of the helicopter to get the car's speed.

(helicopter speed is not mentioned).

car speed = 76.21 - (speed of helicopter).

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