A police car is located 70 ft to the side of a straight road. A red car is driving along the road in the direction of the police car and is 170 feet up the road from the location of the police...

Question

A police car is located 70 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 170 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 70 feet per second. How fast is the red car actually traveling along the road?

The actual speed (along the road) of the red car is _________ feet per second.

Please explain, I'm very lost and I have an exam tomorrow!

Answer 1

Draw a right triangle with vertical side 70 (the distance of the police car from the road),

horizontal side x (the distance of the red car from the vertex of the right angle at time t),

and hypotenuse y (the distance between the red car and police car at time t).

We know that dy/dt = -70.

Find dx/dt when x = 170.

By the Pythagorean Theorem, x² + 70² = y² (**)

So, when x = 170, we have (170)² + (70)² = y². So, y = √33800 = 130√2

Differentiate equation (**) with respect to t:

2x(dx/dt) + 0 = 2y(dy/dt)

So, dx/dt = (y/x)(dy/dt) = (130√2 / 170)(-70) = -910√2 / 17 ≈ -75.7

The speed of the red car is 75.7 ft/sec.