Related Rates
Link the rates of changing quantities with a shared equation.
About this topic
Related rates problems connect the rates of change of different quantities linked by an equation. Draw a diagram, write an equation relating the variables, differentiate with respect to time using the chain rule, then substitute known values. Substitute numbers only after differentiating.
Example: a spherical balloon is inflated at 100 cm³ per second. With V = (4/3)πr³, differentiating gives dV/dt = 4πr² · dr/dt. When r = 5 cm, 100 = 4π(25) · dr/dt, so the radius grows at 1/π, about 0.32 cm per second. Another classic is a 10 ft ladder sliding down a wall: with x² + y² = 100, we get x·dx/dt + y·dy/dt = 0, which links the speed at which the base slides out to the speed the top slides down.
These problems show up in physics, engineering and medicine, for instance how fast a shadow lengthens or how quickly a tumour grows.
Tips: assign variables to quantities that change and constants to those that stay fixed, include units, and mind the signs, since a decreasing quantity has a negative rate. The most common error is plugging in a value like r = 5 before differentiating, which treats a changing quantity as constant.
Key ideas
- Write an equation relating the variables
- Differentiate with respect to time
- Substitute known values
Quick check
Use cases: where Related Rates helps
- Homework: review the key idea behind Related Rates before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Related Rates without notes.
- Visual learning: play with the interactive lab to see how Related Rates changes when you adjust its inputs.
- Teaching a friend: explain Related Rates in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Related Rates?
Link the rates of changing quantities with a shared equation.
What will I learn in Related Rates?
Write an equation relating the variables. Differentiate with respect to time. Substitute known values.
Is Related Rates part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Related Rates lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Related Rates?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.