What Is a Related Rates Problem and How Do You Solve One?

A related rates problem involves two or more quantities that change with time and asks you to find one rate of change from another. You solve it by forming an equation connecting the quantities, differentiating that equation with respect to time, and then substituting the known values.

This is IB Mathematics: Analysis and Approaches HL, sub-topic AHL 5.14: Implicit functions, related rates, optimisation. The method can be examined analytically on Paper 1 or as part of a technology-supported problem on Paper 2.

The Related Rates Method

1. Define the variables and identify the required rate.
2. Write an equation connecting the changing quantities.
3. Use implicit differentiation with respect to time, usually $t$.
4. Substitute the known values only after differentiating.
5. Solve for the required derivative and include units.

The key mechanism is the chain rule. If a quantity $r$ depends on $t$, differentiating $r^3$ gives $3r^2\frac{dr}{dt}$, not merely $3r^2$. Each derivative shows how quickly its quantity changes at the same instant. A positive rate means increasing, while a negative rate means decreasing.

Worked Example

A spherical balloon is being inflated so that its volume increases at $12\pi\text{ cm}^3\text{s}^{-1}$. Find the rate at which its radius is increasing when $r=3\text{ cm}$.

The volume of a sphere is

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

Both $V$ and $r$ depend on time. Differentiate with respect to $t$:

$$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}.$$

Now substitute $\frac{dV}{dt}=12\pi$ and $r=3$:

$$12\pi=4\pi(3)^2\frac{dr}{dt}.$$

Therefore,

$$\frac{dr}{dt}=\frac{12\pi}{36\pi}=\frac{1}{3}\text{ cm s}^{-1}.$$

The radius is increasing at $\frac{1}{3}\text{ cm s}^{-1}$.

Always check that every changing quantity is treated as a function of time, even when $t$ does not appear in the original geometric formula. A labelled sketch can help identify the constraint, but the marks come from the correct equation and differentiation.

Exam Technique

A common misconception is substituting numerical values before differentiating. This destroys the relationship between the changing variables; differentiate first and substitute last. For a command term such as find, examiners expect the linking equation, the differentiated equation, substitution, and a final answer with correct units.