Kim M.

asked • 06/30/17

A police helicopter is hovering. Find the speed of the car.

A police helicopter is hovering at a constant altitude of 0.50 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.

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Arturo O. answered • 06/30/17

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See my comment below for an alternative way to solve this, without having to use calculus.

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Walter B.

Arturo,
The distance from the helicopter to the car, the hypotenuse of the triangle is decreasing at 66 mph, i.e. dz/dt = -66
 
z^2 = x^2 + y^2
 
differentiating wrt  time yields
 
2zdz/dt = 2xdx/dt + 2ydy/dt, we know dz/dt, dy/dt is zero since copter never changes height. Also since y = .5 and z =1, x = (1^2 - .5^2)^.5 =.75^.2 we can solve for dx/dt = -66/.75^.5 or 76 mph
 
 
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06/30/17

Arturo O.

There is a way to solve this without using calculus.  From the geometry of the problem, you have a right triangle of height y, base x, and hypotenuse r.
 
y = 0.5 miles
r = 1 mile
 
The angle from the base to the hypotenuse is θ.
 
θ = sin-1(y/r) = sin-1(0.5/1) = 30°
 
The helicopter measures a component of the car's speed along the hypotenuse, vr.
 
vr = 66 mph
 
But vr is a projection along the hypotenuse of the ground speed vg of the car.
 
vgcosθ = vr
 
vg = vr / cosθ = (66 /cos30°) mph ≅ 76.2 mph
 
The car is moving at 76.2 mph toward the point directly below the helicopter.
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07/01/17

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