Related Rates

As one variable changes, it often causes changes to another variable. With differential calculus seeks to model those changes through related rates. Most of the time, we are trying to see how a variable is changing concerning time.

To compute how the change in one variable causes a change in another you first need to create a fundamental equation or an equation which relates all the variables in question. Then you take the derivative of that function, using the chain rule.

For example, take water flowing into a cylindrical tank, and we want to know how the height of the water in the tank is changing as water flows into the tank at a constant rate.

In this case, the fundamental equation is the volume of a cylinder, because the volume of water in the tank is what’s changing, which is:

Related Rates - 1.png

Drawing a picture of this system gives us the following situation:

Related Rates - 2.png

As the volume of water in the tank changes then the height of the water also changes, but the radius of the water doesn’t because cylinders have a constant radius. So, taking the derivative of the fundamental equation, concerning time is:

Related Rates - 3.png