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# Related Rates

Imagine throwing a stone into a body of water. If the body of water is calm enough, as soon as the stone hits the water, a series of ripples begin to form around the spot the stone went in. The ripples continue to expand outwards as the stone stinks to the bottom. As the ripples expand, the radius of the circular wave grows. As a result, the area enclosed by the ripple also grows.

Framing the problem as a related rate, we could measure the rate at which the enclosed area grows in terms of the rate of change of the radius. Related rates problems are one of the toughest problems for Calculus students to conceptualize. However, this article will further define related rates, how they can be applied in Calculus, and a step-by-step methodology for solving.

## What are Related Rates in Calculus?

With the example of dropping a stone into the water in mind, let's define the term related rates more technically.

Related rates problems typically involve finding the rate at which one variable changes by relating the variable to one or more variables whose rates are known.

In our example of the stone in a calm body of water, as the ripples expand, both the radius and the area enclosed by the wave change. It's important to note the rate at which the radius changes is likely different than that of the area. If we are given one of the rates of change, we can solve for the other rate of change. We can do this because the formula for the area of a circle is related to the radius of the circle.

## The Importance of Related Rates

Solving related rates problems utilizes Calculus skills such as Implicit Differentiation and The Chain Rule. Therefore, make sure to go over our articles before diving into related rates! Learning how to solve these problems will help you strengthen your knowledge of Calculus. Solving related rates of change problems also has extensive applications in finance, physics, travel, and transportation. Framing problems in terms of related rates allows us to write a rate of change in terms of another (typically easier to compute) rate of change.

## Related Rates Formulas

In basic Calculus, related rates problems typically fall into one of two categories:

• Volume or area

• Trigonometry

You'll likely need to brush up on volume/area and geometry formulas you learned years ago. Below are some formulas you might find useful during your work on related rates.

### Area

#### Triangle

• where is base and is height

#### Rectangle or square

• where is height and is width

### Volume

#### Rectangular pyramid

• where is length, is width, and is height

#### Cone

• where is radius and is height

#### Cylinder

• where is radius and is height

#### Rectangular solid

• where is length, is width, and is height

### Trigonometry

Use the diagram below for symbol reference.

SOH-CAH-TOA triangle relationships help in some related rates problems - StudySmarter Original

## Solving Related Rates Problems, Step-by-Step Calculation

Though each related rates problem is different, below is a general methodology for solving related rates of change problems.

### Step 1: Draw a diagram

First and foremost, draw a diagram including what you know and labels of things you need to find.

### Step 2: Identify known and unknown information

Read the problem again to better understand the information the problem provides and what the problem is asking. You can use the important information to label your diagram.

### Step 3: Use equations to relate information in the problem

One of the most challenging parts of solving related rates problems is finding and modeling the relationship between the information you know and the information you are looking for. To relate the two rates of change to each other, think of an equation that involves both variables.

### Step 4: Solve using implicit differentiation

Now that you have an equation, you must perform implicit differentiation on both sides of the equation. You should have an equation in terms of two different rates of change.

### Step 5: Substitute in known values

Finally, you can substitute any information the problem provided. You should be able to solve for one of the rates of change.

Once you've solved a related rates problem, you should always try to interpret what the rate means to ensure your answer makes sense in the context of the problem. You should check both the sign and the magnitude of your answer.

In the related rates problems in AP Calculus, you'll likely need to take the derivative with respect to time when using implicit differentiation.

## Related Rates Problems - Worked Examples

Let's take a look at typical related rates problems. This first one involves a ladder leaning up against a wall.

### Example 1

A 10 ft tall ladder leans against a wall. The base of the ladder begins to slide away from the wall at a rate of 2 ft per second. As the base of the ladder slides away from the wall, the top of the ladder slides down the wall vertically. When the base of the ladder is 9 feet away from the wall, what is the rate at which the top of the ladder slides down the wall?

#### Step 1: Draw a diagram

Drawing a diagram of the problem will help us to better comprehend our known and unknown values.

From the horizontal rate of change, we are tasked with finding the vertical rate of change - StudySmarter Original

Based on our diagram, we are missing the vertical rate of change. However, we do have the horizontal rate of change and the length of the ladder.

#### Step 2: Identify known and unknown quantities

Before we can do any Calculus, we must fully understand the problem. We know that a 10 ft ladder slides away from a wall horizontally at a rate of 2 ft per second. The problem wants to know at what rate the top of the ladder moves when the base of the ladder is 9 ft from the wall.

Using our diagram in step 1, we can organize known and unknown variable quantities:

are functions of time in this problem, so they are written . However, the length of the ladder, , does not change with time, so it is not written with function notation.

#### Step 3: Use equations to relate information in the problem

Based on the information we have and the information we need, it should be obvious that the Pythagorean Theorem will be useful in this problem.

Looking at the diagram again, notice that the ladder and 2 walls make a right triangle. This is a perfect scenario to use the Pythagorean Theorem!

Remember, while the ladder moves horizontally and vertically, the hypotenuse of the triangle (length of the ladder) does not change.

Notice that we are given the derivative of x with respect to time,

We are also asked to find the rate at which the ladder is moving vertically,

How can we make an equation with these variables? Implicit differentiation!

#### Step 4: Solve using implicit differentiation

Now that we have an equation, let's use implicit differentiation to get the equation in terms of two rates of change. We will take the derivative with respect to time.

#### Step 5: Substitute in known values

Again, we want to find the rate at which the ladder slides down the wall vertically:

We know that ft and

Plugging in our known values, we get

To solve for , we still need the value of when . We can use the Pythagorean Theorem equation we set up earlier to find , subbing .

Plugging in and solving for

The negative sign in our answer signifies that the ladder moves in the negative direction (downwards). Therefore, the ladder slides down the wall at a rate of 4.129 ft per second when the base ladder is 9 ft from the wall. Considering the ladder moves at a horizontal rate of 2 ft per second, the magnitude of our answer also makes sense!

Considering a perfectly spherical balloon being filled with air. The balloon expands at a rate of per second. When the balloon's radius is 4 cm, how fast is the radius increasing?

#### Step 1: Draw a diagram

From the rate of change of the volume, we are tasked with finding the rate of change of the radius - StudySmarter Original

Based on our diagram, we are missing the rate of change of the radius. However, we do have the rate of change of the volume.

#### Step 2: Identify known and unknown quantities

We know that the volume of a spherical balloon increases at a rate of per second. We want to know the rate of change of the radius when the balloon has a radius of 4 cm. Organizing into variables, we have

#### Step 3: Use equations to relate information in the problem

Based on the information we need, and the shape of the balloon, the volume equation of a sphere will be useful in this problem.

#### Step 4: Solve using implicit differentiation

Now that we have an equation, let's use implicit differentiation to get the equation in terms of two rates of change. We will take the derivative with respect to time.

#### Step 5: Substitute in known values

We want to find the rate of change of the radius:

We know that and

Plugging in our known values, we get

Therefore, the radius expands at a rate of about . Clearly, the radius grows at a very slow rate. However, this makes sense considering the volume also grows at a relatively slow rate.

## Related Rates - Key takeaways

• Related rates problems typically involve finding the rate at which one variable changes by relating the variable to one or more variables whose rates are known.
• Solving related rates problems allows us to write a rate of change in terms of another (typically easier to compute) rate of change.
• Though each related rates problem is different, a general methodology for solving is:
• Identify known and unknown quantities
• Draw a diagram
• Use equations to relate information in the problem
• Solve using implicit differentiation (with respect to time)
• Substitute in known values

To calculated a related rate, find an equation that highlights the relationship between the known rate of change and the unknown rate of change. Then, use implicit differentiation (usually with respect to time) to solve.

Related rates problems typically involve finding the rate at which one variable changes by relating the variable to one or more variables whose rates are known.

Important formulas for related rates include area and volume equations and geometric equations (including SOH-CAH-TOA and the Pythagorean Theorem).

Consider an expanding balloon. Obviously, the volume of the balloon increases. However, the radius of the balloon also increases. The volume equation of a sphere relies on the length of the radius. Thus, the rate at which the volume of the balloon changes is related to the rate at which the radius changes.

Related rates have extensive applications in finance, physics, and travel and transportation (among other fields). Framing problems in terms of related rates allows us to write a rate of change in terms of another (typically easier to computer) rate of change.

## Final Related Rates Quiz

Question

What are related rates problems?

Related rates problems typically involve finding the rate at which one variable changes by relating the variable to one or more variables whose rates are known

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Question

What Calculus skills do related rates problems rely on?

• Implicit differentiation
• Chain rule

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Question

Think of an example of a related rates problem.

Consider a trough filled with water. There is a hole at the bottom of the trough that the water flows out through. As the water flows out, the level of the water in the trough decreases. The rate of change of the outflow of water is related to the rate of change of the water level.

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Question

Consider the trough of water example mentioned. When differentiating your equation, you will take the derivative of each side with respect to _________.

time

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Question

What is the first thing you should always do when solving a related rates problem?

Draw a diagram!

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Question

In AP Calculus, related rates problems fall into one of two categories:

Volume/area or trigonometry

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Question

Why is framing a problem in terms of related rates useful?

Solving related rates problems allows us to write a rate of change in terms of another (typically easier to compute) rate of change

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Question

What is a general process for solving relate rates problems?

To calculated a related rate, find an equation that highlights the relationship between the known rate of change and the unknown rate of change. Then, use implicit differentiation (usually with respect to time) to solve.

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Question

Is there a related rates formula?

Not necessarily. However, there are important formulas to keep in mind when solving related rates problems, like formulas for area and volume as well as general trigonometry formulas.

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Question

Consider a ladder that slides down a wall. Which equation might be useful when calculating the rate of change of the ladder's movement in the direction?

The Pythagorean Theorem!

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