Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Related Rates

Differentiate a geometric/physical constraint with respect to time and connect simultaneous rates of change.

AA HL · AHL 5.14

Learning goal

Solve related-rates problems with correct variable definitions, signs and units.

Syllabus link

AA HL: AHL 5.14 · Topic 5 Calculus.

Big idea

The variables change together because a constraint links them. Differentiate the constraint with respect to time.

Key relationship

\(\frac{d}{dt}F(x(t),y(t),\ldots)=0\)

1

Core idea

The variables change together because a constraint links them. Differentiate the constraint with respect to time.

Solve related-rates problems with correct variable definitions, signs and units.

2

Key results and method

\[\frac{d}{dt}F(x(t),y(t),\ldots)=0\]
  1. Draw and label the geometry.
  2. Treat changing quantities as functions of time.
  3. Write a relation between them.
  4. Differentiate with respect to time before substituting numerical values.
  5. Interpret sign and units.
3

Worked example

Question

A sphere has volume \(V=\frac43\pi r^3\). If \(dV/dt=12\pi\) when \(r=2\), find \(dr/dt\).

Solution
  1. \(dV/dt=4\pi r^2dr/dt\).
  2. \(12\pi=4\pi(2)^2dr/dt\).
Answer: \(dr/dt=3/4\) units per unit time.
4

Practice

Try these questions before moving on.

  1. Solve a sliding-ladder problem.
  2. Relate the changing height and radius of a cone.
  3. Explain why a negative rate can be the correct answer.

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5

Tutorials and premium resources

▶ Related rates full lesson
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