Topic 4.5 Notes – Solving Related Rates Problems

1. What Related Rates Problems Are

A related rates problem asks you to find how fast one quantity is changing by using information about how another quantity is changing.

The structure is always the same:

• Several variables (like $x, y, r, V, A$) are changing.
• They are connected by an equation.
• Every variable depends on time $t$, even if $t$ is never written.
• You are given at least one rate (like $\frac{dx}{dt}$).
• You must find another rate (like $\frac{dy}{dt}$) at a specific moment.

Big idea:
You are not differentiating with respect to $x$. You are differentiating with respect to time.

That’s why implicit differentiation and the chain rule show up every time.

2. The Equations That Connect the Variables

Before you can differentiate anything, you need an equation that connects the quantities. Most come from geometry.

Here are the ones that appear constantly:

$$A = lw$$

$$A = \pi r^2$$

$$V = \pi r^2 h$$

$$V = \frac{4}{3}\pi r^3$$

$$x^2 + y^2 = z^2$$

Two important reminders:

• If something is constant, its derivative is 0.
• Do not plug numbers in before differentiating unless the quantity is actually constant.

For example, if a rope has fixed length 20 meters, then
$$\frac{d}{dt}(20) = 0$$

That fact usually simplifies the equation a lot.

3. The Method Every Time

The problems look different. The method does not.

Step 1. Identify the rates

Write given information as derivatives.

Instead of “the radius increases at 3 cm/sec,” write
$$\frac{dr}{dt} = 3$$

Be clear about what you’re solving for.

Step 2. Draw and label a diagram

For something like a ladder sliding down a wall, your sketch might look like the left part of the diagram below:

Sliding ladder related rates setup

Label every changing quantity with a variable. This step prevents mixing up what’s constant and what’s changing.

Step 3. Write an equation relating the variables

Using the ladder example:

$$x^2 + y^2 = L^2$$

This equation connects the quantities. Without it, there’s nothing to differentiate.

Step 4. Differentiate with respect to time

This is where most mistakes happen.

If
$$x^2 + y^2 = L^2$$

then

$$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2L\frac{dL}{dt}$$

Every variable picks up a derivative. Every time.

If $L$ is constant, then $\frac{dL}{dt} = 0$.

Students often forget the chain rule and write $2x + 2y$. That will cost points immediately on an FRQ.

Step 5. Plug in values at the specific moment

Only after differentiating do you substitute:

• numerical values
• known rates
• the specific time or position given

Sometimes you must first find a missing value (like solving for $y$ using the original equation).

Step 6. Solve and interpret

Solve algebraically for the unknown rate.

Then interpret:

• Positive → increasing
• Negative → decreasing

If $\frac{dy}{dt} = -4$, don’t just stop there. Say:
“The height is decreasing at 4 meters per second.”

Units must match the quantity. Area should give square units per time. Volume should give cubic units per time. If your units don’t make sense, something went wrong.

4. Interpreting Related Rates in Context

AP questions care about meaning, not just computation.

You may be asked:

• Is the volume increasing or decreasing?
• Is the distance between objects growing faster or slower over time?
• How fast is something shrinking?

Always connect your answer back to the physical situation.

Also watch for these common traps:

• Plugging numbers in before differentiating
• Forgetting a chain rule factor
• Forgetting that a constant’s derivative is 0
• Not solving for missing geometric values first

On free-response questions, missing the chain rule usually loses multiple points at once. On multiple choice, it leads to answers that look tempting but are missing a factor.