How fast is the red car actually traveling along the road?

Question

A police car is located 50 feet to the side of a straight road.

A red car is driving along the road in the direction of the police car and is 120 feet up the road from the location of the police car. The police radar reads that the distance between the police car and the red car is decreasing at a rate of 85 feet per second. How fast is the red car actually traveling along the road?

Answer 1

Form an x-y axis. Put the police car 50 feet east of the road and the red car 120 feet north on the road.

This forms a right triangle with the distance between the police car and red car being the hypotenuse.

Let s be the distance:

s² = y² + 50²

ds/dt = -85 ft/sec as given

Taking the derivative of both sides of the distance equation:

2s ds/dt = 2y dy/dt

We want to find dy/dt when y is 120 feet.

dy/dt = (s/y)ds/dt

s = √(120² + 50²) = 130 feet

dy/dt = (130/120)(-85) ≈ -92.08 ft/sec or 92.08 ft/sec going south. (≈62.78 mph)

Answer 2

Draw a diagram of a right triangle with one leg labeled 50 and the other labeled x. Label the hypotenuse h (or something.

There is a difference between the rate at which the hypotenuse is shrinking (which is the measured - 85 ft/sec, dh/dt) and the rate at which the leg is shrinking (dx/dt , which is the speed we want).

Fortunately, we can write the equation that models the relationship between x and h, differentiate with respect to t, and solve for dx/dt:

x² + 50² = h²

2x·dx/dt = 2h·dh/dt. At the instant that x = 120 , h = 130. Plug those values in to get the desired speed.