This chapter deals with two different types of word problems that involve motion: related rates and the
relationship between velocity and acceleration of a particle. The subject matter might seem arcane, but
once you get the hang of them, you’ll see that these aren’t so hard, either. Besides, the AP Exam tests only
a few basic problem types.
RELATED RATES
The idea behind these problems is very simple. In a typical problem, you’ll be given an equation relating
two or more variables. These variables will change with respect to time, and you’ll use derivatives to
determine how the rates of change are related. (Hence the name: related rates.) Sounds easy, doesn’t it?
Example 1: A circular pool of water is expanding at the rate of 16π in.^2 /sec. At what rate is the radius
expanding when the radius is 4 inches?
Note: The pool is expanding in square inches per second. We’ve been given the rate that the area is
changing, and we need to find the rate of change of the radius. What equation relates the area of a circle to
its radius? A = πr^2.
Step 1: Set up the equation and take the derivative of this equation with respect to t (time).
= 2πrIn this equation, represents the rate at which the area is changing, and is the rate at which the
radius is changing. The simplest way to explain this is that whenever you have a variable in an equation
(r, for example), the derivative with respect to time represents the rate at which that variable is
increasing or decreasing.
Step 2: Now we can plug in the values for the rate of change of the area and for the radius. (Never plug in
the values until after you have taken the derivative or you will get nonsense!)
16 π = 2π(4)Solving for , we get
16 π = 8π and = 2The radius is changing at a rate of 2 in./sec. It’s important to note that this is the rate only when the radius