Key takeaway: Related rates problems use an equation that links changing quantities, then differentiate it with respect to time to find an unknown rate from a rate that is given.
Grade 12 – AP Calculus ABAP Calculus AB Unit 4Calculus
Definition of Related Rates
In related rates problems, two or more quantities change with time and are tied together by a geometric or physical equation. Once one rate is known, the equation forces the other rate at that same moment.
The routine never varies. Sketch and label, write one equation among the variables, take the derivative of each side with respect to time \(t\) using the chain rule, and only afterward put in the numbers for the instant asked about.
Formula and notation
Rates are derivatives with respect to time, such as \(\frac{dr}{dt}\) and \(\frac{dA}{dt}\). For a circle of area \(A=\pi r^2\):
\[ \frac{dA}{dt}=\frac{dr}{dt}\cdot 2\pi r. \]
A negative sign on a rate signals shrinking.
Key facts
- Put in instant values only after differentiating; before that, the variables are still moving.
- Fixed quantities, such as a tank radius, disappear when differentiated.
- Related rates always start with a labeled sketch and one named letter per changing quantity.
- Check the units of the answer: they should be a quantity unit divided by time.
Related Rates example
The edge of an oil slick on a calm pond is a circle, and its radius increases by 0.4 feet each minute. At the instant when the radius is 15 feet, how quickly is the covered area increasing?
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Choose the link between the variables: \(A=\pi r^2\).
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Take the time derivative of each side, giving \(\frac{dr}{dt}\) times \(2\pi r\) on the right.
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Only now substitute \(r=15\) and \(\frac{dr}{dt}=0.4\): \(2\pi(15)(0.4)=12\pi\).
The area is increasing by \(12\pi\approx\)37.7 square feet per minute.
Common mistake
Watch out
A student writes that the rate is \(30\pi\), from \(2\pi\cdot 15\), differentiating \(\pi r^2\) as if \(r\) were the independent variable.
Why it is wrong
Time is the independent variable here, so the chain rule adds the factor \(\frac{dr}{dt}\). Without it, the units come out as feet instead of square feet per minute. Real related rates answers always carry the rate of the given quantity; including the 0.4 gives \(12\pi\), not \(30\pi\).
Related terms and lessons
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