4-C Related Rates (Circle Example)
Instructions
Start by playing the applet (the play button is in the bottom-left corner of the graphic) and observing the growing circle on the left. Our goal is to examine the rate of change in the circle's area with respect to time.
- Use the slider tool or input box for to set a specific time.
- Use the slider tool or input box for to set an increment (small change) in time.
- The variables time, radius, and area create a chain or composition: . Observe the three different rates of change involved in relating these three variables.
- The graph on the right shows the area of the circle as a function of its radius, and the radius is a function of time. Use the input box for r(x) to set how the radius changes over time. (Note that x is being used in place of t because the calculator requires x as the input variable.)
4-C Related Rates of Change
Any science or math formula that you can think of typically relates two or more quantities to one another.
For example, kinetic energy (energy due to motion) is given by . This relationship allows us to analyze how changes in velocity can lead to changes in energy. Naturally, we could describe this with a derivative (of energy with respect to velocity):
(Fun fact: momentum is mass times velocity.) But what happens if the velocity of the moving object is also changing as a function of time? Then, the kinetic energy of the object is a function of velocity, which is a function of time. This creates a chain/composition: , and we can replace the variables in the equation with functions of time:
Now, not only can we ask how the kinetic energy changes with respect to velocity, we can also ask how the kinetic energy changes with respect to time. These are two different rates of change with different units of measurement and different interpretations. How do we differentiate in the above equation (with respect to t) without knowing what v(t) is? For example, v(t) could be , or , or .
The premise of related rates problems is that we start with a relationship (equation) between two (or more) quantities and then use implicit differentiation (usually with respect to an implicit variable for time, which may not even be present in the equation) to find a relationship (equation) between the rates of change of these quantities.