Related Rates

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Today’s plan

[greybox2]

  • Overview
  • Warm-up
  • Lecture problems
  • Midterm return [/greybox2]

Overview

Related rates is notorious among many of being one of the more difficult types of problems to solve because:

  1. Word problems.
  2. Implicit differentiation.

These problems become much more natural to approach with consistent practice, but they tend to be tricky when you start doing them for the first time.

Dr. Browne’s 5-step process is good outline for helping you figure out what the pattern is to these kinds of problems. Therefore, I will give you a copy of it here:

[greybox2] Dr. Browne’s 5-step process for Related Rates:

  1. If applicable, draw a picture and give letters for variables.
    • Specify what your variables represent CLEARLY. A LOT of confusion regarding word problems arises from not being specific enough when defining variables.
  2. Determine values you know and what you need to find.
    • Values describing change (also includes velocity) typically correspond to derivatives.
  3. Find an equation involving our variables.
  4. Use implicit differentiation.
  5. Replace variables with values from step 2 and solve for the unknown. [/greybox2]

The two problems we will work on today are related rates problems that can be done with the above 5-step process. Before we jump into the problems, I want to point out a couple of things that will help you in the problems as a “warm-up”.

Warm-up

[greybox2] Warm-up 1. Google what “North”, “South”, “West”, and “East” mean, even if you know them. Many points have been lost over making silly mistakes by confusing which direction is which.

Please do not use AI (such as ChatGPT or DeepSeek) for this. It’s 2025 and you should know how to use search engine to research information before tackling a problem. [/greybox2]

[greybox2] Warm-up 2. Suppose we have a car that can only go left or right. Let $x$ be the (position) of the car and assume that $x$ depends on time $t$. Answer the following:

  1. What does it mean for $x$ to be negative?
  2. What does $\dfrac{dx}{dt}$ physically represent?
  3. What does it (physically) mean for the car if $\dfrac{dx}{dt}$ is positive? Negative? Zero?
  4. Can $x$ be negative and $\dfrac{dx}{dt}$ be positive at the same time? Explain. [/greybox2]

[greybox2] Warm-up 3. Look up the statement of the Pythagorean theorem and keep it on the side. You may find it helpful later.

Please do not use AI (such as ChatGPT or DeepSeek) for this. It’s 2025 and you should know how to use search engine to research information before tackling a problem. [/greybox2]

Lecture Problems

[greybox] Problem 2 from Section 3.6. Two cars start from the same point. Car 1 starts at time 0 and is traveling at a velocity of 20mph east. Car 2 starts at time 1 and is traveling at a velocity of 30mph north. Find the rate at which the distance between the two cars is changing after 5 hours. [/greybox]

[hint]

  • Calculate the distance the two cars have traveled by using
\[\begin{align*} \text{distance} = \text{speed} \times \text{time}. \end{align*}\]
  • Car 1 has traveled for 5 hours and Car 2 has traveled for $4$ hours. Use the Pythagorean theorem to set up your equation and then apply implicit differentiation. [/hint]

[solution] $38.85$mph [/solution]

[greybox] Problem 4 from Section 3.6. A $10$ft ladder is leaning against a wall over a flat floor. The ladder begins to fall down the wall at $0.5$ft/sec. How fast is the ladder sliding away from the wall when the base is $4$ft from the wall? [/greybox]

[hint] If the ladder is $10$ft and leaning against a wall, we get a triangle. Apply the Pythagorean theorem to set up your equation and then apply implicit differentiation. [/hint]

[solution] $1.146$ft/sec. [/solution]

Midterm return

If you were not in class on Wednesday when I passed out the exams, please check the instruction I put on the implicit differentiation page. I will pass them out again today and give you some time to look over them.