12.5 Related Rates
Equations relate two or more quantities (represented by variables). Sometimes,
we are interested in how these quantities are changing, not with respect to each
other, but with respect to a completely different quantity that is not included in
the original equation (which will be time for the examples we will encounter). If
we are given enough information, including the rate of change for one of these
quantities with respect to time, we can then solve for the other related rate, also
with respect to time.
Example #1: A circular ripple in the water is growing. The area is growing at a
rate of 5 square inches per second. At the moment that the Area is 30 square
inches, how fast is the radius growing?
The first thing we need to solve is the equation that relates the quantities given in
the problem. In this case, the quantities referred to are the Area of a circle (in
square inches) and the radius of the circle (in inches). This leads us to the Area
formula of a circle,
𝐴 = 𝜋𝑟
2
.