Related rates problems combine derivatives, word problems, and real-world applications — making them a favorite for the AP Calculus AB & BC exams. These appear in both multiple-choice and free-response questions, often worth 4–6 points on FRQs.
In this RevisionDojo guide, you’ll learn:
- How to recognize a related rates problem
- The 6-step method to solve them
- An AP-style worked example
- Common mistakes that cost points
📚 What Are Related Rates Problems?
These problems involve two or more variables that change over time. You’ll be given the rate of change of one quantity (dxdt\frac{dx}{dt}, for example) and asked to find another rate at a specific instant.
Example topics:
- Water flowing into or out of a container
- Moving objects (boats, cars, planes)
- Expanding/shrinking geometric shapes
🔍 Step-by-Step Strategy
1. Read Carefully & Identify Quantities
Underline:
- The given rates
- The quantity you need to find
2. Draw a Diagram (If Possible)
Helps you visualize relationships between variables.
3. Write the Relationship Equation
This should connect all variables — often a geometry or physics equation.
4. Differentiate Implicitly with Respect to Time
Apply chain rule since variables are functions of tt.
5. Substitute Known Values
Only plug in numbers after differentiating. This avoids losing variables you need.
6. Solve for the Unknown Rate
Make sure to include units in your final answer.
📝 Example AP-Style Problem
Problem:
A spherical balloon is being inflated so that its volume increases at a rate of 100 cm3/s100 \, \text{cm}^3/\text{s}. How fast is the radius increasing when the radius is 5 cm5 \, \text{cm}?
Solution:
- Given: dVdt=100\frac{dV}{dt} = 100, r=5r = 5, find drdt\frac{dr}{dt}
- Volume equation: V=43πr3V = \frac{4}{3} \pi r^3
- Differentiate:
dVdt=4πr2drdt\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}
- Substitute:
100=4π(5)2drdt100 = 4\pi (5)^2 \frac{dr}{dt}
- Solve:
100=100πdrdt⇒drdt=1π cm/s100 = 100\pi \frac{dr}{dt} \quad \Rightarrow \quad \frac{dr}{dt} = \frac{1}{\pi} \, \text{cm/s}
⚠️ Common Mistakes to Avoid
- Plugging in values before differentiating (loses the general relationship)
- Forgetting to apply the chain rule for squared or cubed terms
- Dropping units in the final answer
- Confusing sign of rates (e.g., radius shrinking means negative rate)
📊 Practice Strategy from RevisionDojo
- Drill at least 1 related rates problem every few days
- Practice with diagrams until you can quickly identify the right equation
- Use past AP FRQs — College Board frequently reuses similar problem styles
🧭 Final Advice from RevisionDojo
Related rates problems can look intimidating, but they all boil down to one equation, one derivative, and one substitution. Once you follow the process, they become some of the most predictable FRQ questions you’ll face.
