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AP Calculus Review: Related Rates

Related Rates problems are any problems where we are relating the rates of two (or more) variables. They come up on many AP Calculus tests and are an extremely common use of calculus.

AP Calculus Review: Related Rates

There is a series of steps that generally point us in the direction of a solution to related rates problems. No two problems are exactly the same, but these steps are a very good rubric for solving a wide variety of problems:

1. We must first identify the variables which are changing in the problem. This could be size, volume, distance, etc.
2. Find the governing equation which relates the variables. This is often given in the problem, or is a relatively well-known relation (i.e., volume = length × width)
3. Rates are usually (for AP Calculus) in relation to time. Therefore, we differentiate both sides with respect to time.
4. Substitute in the known rates of change and/or known quantities
5. Solve for wanted rate of change or quantity

Common Errors:

We always plug in known values of variables after finding the derivative, never before finding the derivative.

A difficult Related Rates example.

(Note: this is a more difficult than normal problems in its set up. We have simpler sample problems following.)

A ladder with a length of 1 meter is leaned up against a building. It’s a bit off balance, and so beings to slide away from the building at a rate of r m/s. What is the rate that the top of the ladder moves while sliding down the building when the base of the ladder is b meters from the building?

1. Variables: what changes with time?
a. Speed of the top of the ladder
b. Distance of base from the building

2. Governing equation:
a. Drawing this problem makes it easy to visualize.

Related Rates

b. The moment we see a right triangle, we use the Pythagorean theorem
c.screen-shot-2016-12-30-at-1-19-16-pm
d. we are looking for dy/dt

3. Differentiate with respect to t:

screen-shot-2016-12-30-at-1-19-26-pm

We should automatically see at this point that since l is constant, dl/dt is 0. I will leave it for a few steps for purpose of demonstration

4. We can now substitute in what we have.
a. We know dx/dt = r, dl/dt = 0, and x = b

screen-shot-2016-12-30-at-1-19-34-pm

screen-shot-2016-12-30-at-1-19-39-pm

b. We know also know y from the original Pythagorean theorem.

screen-shot-2016-12-30-at-1-19-45-pmwhich are all known quantities.

screen-shot-2016-12-30-at-1-19-49-pm

Related Rate Example Problems

Volume questions are quite common examples of Related Rate Problems.

A spherical balloon is being inflated so that its volume increases at a rate of 20 cm3/s. How fast is the radius of the balloon changing when the radius is 10 cm?

Solution:

1. Identify Variables:
Volume
Radius

2. Governing Equations

screen-shot-2016-12-30-at-1-19-55-pm

3. We differentiate both sides with respect to time.

screen-shot-2016-12-30-at-1-32-10-pm

4. Substitute in the known rates quantities

screen-shot-2016-12-30-at-1-20-47-pm

5. Solve for wanted rate of change or quantity

screen-shot-2016-12-30-at-1-20-59-pm

Another similar practice problem:

A spherical balloon’s area is increasing at the constant rate of 5 cm/sec. How fast is the volume increasing when the radius is 10 cm?

Solution:
This is a two-part question.
First we compute change in r, and then computer the equivalent change in V.

Let’s use the same series of steps as before:

Part 1:

 

Part 2:

screen-shot-2016-12-30-at-1-21-18-pm

 

 

 

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About Zachary

Zachary is a former mechanical engineer and current high school physics, math, and computer science teacher. He graduated from McGill University in 2011 and spent time in the automotive industry in Detroit before moving into education. He has been teaching and tutoring for the past five years, but you can also find him adventuring, reading, rock climbing, and traveling whenever the opportunity arises.


4 Responses to “AP Calculus Review: Related Rates”

  1. William Xavier Vebert says:

    I’m pretty sure you have the surface area of a sphere wrong.

    • Magoosh Test Prep Expert Magoosh Test Prep Expert says:

      Hi William,

      We are dealing with the volume of a sphere, not the surface area. Is this the discrepancy you see?

  2. Glen says:

    In part 1 of your last problem you give the surface area as A = 4pi*r^3. It should be A = 4pi*r^2. Also in the statement of the problem you have the units for the area as cm/sec though you correct it below to cm^2/sec.

    • Magoosh Test Prep Expert Magoosh Test Prep Expert says:

      Hi Glen,

      Great catch! Thanks for bringing this to our attention–we changed the equations for the last problem, so it should be correct now 🙂

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