Related Rates

Related Rates

by Daniel Kotovsky

What are related rates?

Related Rates Explained Simply

A related rates problem is a common type of problem in calculus involving two or more related variables changing at the same time. The idea is that if you know the rate of change of one variable, you should be able to find the rate of change of the other using calculus.

Example Problem

Riley and Simon

Riley said something disturbing to Simon, which caused Simon to run away from Riley at a rate of 5 meters per second. Imagine the distance between Riley and Simon as the radius of a circle. Assuming Riley remains stationary, at what rate is the area of this imaginary circle changing?

Multiple Choice

How do you express a changing radius with respect to time?

• dr/dx

• dr/dt

• dx/dy

• dy/dt

Correct!

In calculus, variables that are constantly changing are expressed with a derivative, and the expression (A) "with respect to" (B) essentially just means (A)/(B).

That means that in this case, where you have a changing radius (r) with respect to a changing time (t), you would express that as dr/dt.

We're trying to find a changing area (A) with respect to the changing time (t). Knowing what we know, we can express that as dA/dt.

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​What are we trying to find?

Identifying Variables

dA/dt = ?

​What We're Finding

r = 10m

dr/dt = 5m/s

A = πr^2 = π(10)^2 = ~314.159 m^2

​What We Know

The hard part...

You can get a clue from the name of the concept we're working with here. They're called "related" rates, so first we want to find how the radius of a circle relates to its area.

Multiple Choice

What is the relationship between the radius of a circle and its area?

• $$A\ =\ \pi r^2$$

• $$A\ =\ 2\pi r$$

• $$A\ =\ \frac{r}{2\pi}$$

• $$A\ =\ r+\pi$$

Well done!

Now that we know how area (A) and radius (r) are related, we can use the chain rule to take the derivative with respect to time of both sides.

​You've solved the problem!

If you still don't understand how to do related rates problems, that's fine, because related rates are very difficult and take a lot of practice. However, I hope that this has at least helped you to see the fundamentals of how it works and why it works.