Race car, rate of change of distance: A spectator is at a car race. When a car is approaching point A at 120 mph = 176 ft/sec, and is 100 ft from A, how fast is the distance, h, between the spectator and the car changing?

Using the diagram, let x be the position of the car, and . It is negative because the car is approaching point A and x is getting smaller. h is the distance between the car and the spectator. It is related to x via the Pythagorean Theorem:

, and

We wish to find the rate of change of h with respect to time as the car moves. Using the relationship between x and h, we find

We now substitute the values of x and :