Related Rates

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We discuss how to solve real world style problems that involve interrelated rates of change.

Video Lecture

Text and Additional Details

Whenever we have a problem that involves some kind of rate of change, chances are excellent that calculus is lurking somewhere in the situation... like some kind of horror movie killer waiting for you to strike out on your own. But, often we don’t have all the primary information we need, rather one rate of change is somehow based on some other loosely related rate of change. When you have one rate dependent on another, this is generally referred to as a “related rates” problem, and that’s what we will dive into here.

You may be wondering why I’m being so vague, but the truth is that related rates problems are very general and can encompass all kinds of different scenarios, contexts, setups, and equations. There is no single formula or even type of formula to solve a related rates problem. Instead, we will cover a general framework that should help you determine how to approach and solve a related rates problem, but keep in mind that every such problem is different and so is the process to solve them. So although we cover an algorithmic approach here, not every piece will apply to every problem.

1 : Related rates problems are problems that...

Involve two or more rates that are related to one another. Involve determining a rate of change. Require an approach unlike any other problems. Cause the most students to fail calculus.

Lets start with an example. Consider the following problem:

2 : Water from a water heater is leaking onto the basement floor of a home. A circular puddle is created whose area is increasing at the rate of 25 square inches per minute. How fast is the radius of the puddle increasing when it (the radius) is 10 inches?

We outline a five step process to tackle these kinds of problems here -but again- note that not all steps are always necessary, and sometimes there are other things you need to consider. This is a general framework with which to approach these kinds of problems, but every problem is a little different.

Step 1:: The first step, as is usually the case in most math problems, is to try and draw out a quick sketch of what you are dealing with. The sketch doesn’t have to be accurate - the purpose is not to be used to solve the problem directly, but rather as a way to get the information you are given down into a structured context that helps you relate each piece of information together. Even if the problem doesn’t have anything visual, it helps to write down the key words or ideas and some arrows/formulas/inequalities/relationships between them so that you can literally see how the different parts of the problem are related to one another.
In the case of our current problem, we’ll start with a circular puddle.
Step 2:: The second step is to apply mathematical language to everything in the problem that we can. It’s important to keep in mind that it is easier to cut back later than to make assumptions beforehand and have to fix them later. To this end, it’s a good idea to assign a letter to everything in the problem that might be useful, even if it is something that we know a number for. This is the most frequently made mistake by students, assuming that since they have a number for something it doesn’t need a variable. Again, it’s best to go too general now and cut it back later.
For our problem, we have a puddle forming on the floor and we want to know something about the radius of that puddle, so we can mark the radius of the puddle with $r$ . The area is also mentioned, so we will denote that by $A$ . Finally, time is an obvious element (we’re talking about the puddle expanding over time) so we will denote time in minutes (since that was the unit mentioned) by $t$ . Notice that we also have a rate given, which is the increase in area over time (the “25 square inches per minute”). This represents a change in area over time, and since we’ve denoted area by $A$ and time by $t$ , this would be represented by $\frac {dA}{dt}$ . Notice that, although we “know” the radius (meaning we have a value for the radius of “10” mentioned in the problem) and we were given a value for $\frac {dA}{dt}$ , we aren’t making any assumptions yet, we are leaving all these values as variables. Keep it general now, substitute stuff later.

3 : When first applying mathematical language to the problem, it is best to...
Start out with as few variables as possible - you can always add more later, and having fewer is easier to keep track of. Start out by assigning variables to as many things as seem possibly relevant - you can cut them down again later, but this way you won’t miss something while setting up equations. Keep track of as many different possible variables as possible, but don’t worry about assigning them until later. Just remember that they might need it. Cross out as much of the problem as you can get away with, until it’s really just an algebra 2 problem.
Step 3:: The third step is to write down an equation that relates the stuff we know about, i.e. the variables we’ve come up with. This can be tricky as there may be a number of options, or no obvious options at all. Unfortunately there’s no systematic way to determine the “right” formula, this is the difficult part of related rates. Typically you want to try to find a formula that includes the variable you want to find a rate for, and variables that you already know information about - although as mentioned this isn’t always straight forward.
For our example, we have area and radius, and we know the puddle is a circle, so it isn’t too much of a stretch to start with the typical area formula for a circle: $A = \pi r^2$ . This is a good place to start, but remember that it may be necessary to come back and try a new function if the first one you try doesn’t end up having what you need.
Step 4:: Now we want to figure out a way to generate the rate we want, probably along with any rates we were given. This means we want to take a derivative, (almost but not always!) with respect to time. This will inevitably involve implicit differentiation since we will have multiple variables and not all of them will match the differential variable.
For our puddle problem we want to know the rate at which the radius is increasing with respect to time, so (using our notation) we want to find $\frac {dr}{dt}$ . With this in mind, let’s take the derivative with respect to time of our equation $A = \pi r^2$ . Doing this will get us $\frac {dA}{dt} = 2\pi r \frac {dr}{dt}$ . Notice here that we are using the $\frac {dA}{dt}$ style form rather than $A'$ form for the derivative. With so many different variables floating around, this notation is extremely helpful to keep things straight.
Step 5:: Step five is to substitute in whatever we can into the result we got from taking the derivative in the previous step. We may end up needing more information, but that doesn’t necessarily mean the formula we chose in step three was the wrong one. If there are missing pieces of information, first try and see if you can find them or figure them out another way before scrapping everything and trying a new formula.
For our puddle problem, we have $\frac {dA}{dt} = 2\pi r \frac {dr}{dt}$ , and we know that $\frac {dA}{dt} = 25$ from the original statement. We also know that, at the time we are interested, $r = 10$ . So plugging these in we get $25 = 2\pi (10) \frac {dr}{dt}$ . Solving for $\frac {dr}{dt}$ we get $\frac {dr}{dt} = \frac {5}{4\pi }$ inches per minute.

Notice, if we hadn’t left things generalized, and plugged in numbers too early, we could have had an equation that looked like $A = \pi (10)^2$ (if we plugged in the $r=10$ value at the start). But then, when we tried to take a derivative of both sides we would have gotten $\frac {dA}{dt} = 0$ . But, not only is that not helpful (there’s no $\frac {dr}{dt}$ to be found), it’s actually wrong since we know $\frac {dA}{dt} = 25$ . If you run into a situation like this, it’s a definite red flag that something went awry earlier in your formula process, and you probably plugged in a value as something you “knew” rather than leaving it as a variable until after you took the derivative. A good rule of thumb: The formula you take a derivative of, almost always is going to be the general form of that formula before anything is plugged in. Formulas like the Pythagorean Theorem ( $A^2 + B^2 = C^2$ ) or formulas for areas or volumes of shapes for example - not those formulas with parts “plugged in” beforehand.

We have introduced the idea of related rates, and went through an example using a five step algorithmic approach. However, this algorithmic approach is more of a series of general guidelines to help navigate related rate problems, but each problem is a little different from all the other problems, so it is a good idea to practice them as much as you can, and watch the example videos for more examples on how to do these problems.