Exponential models for real-world decay (IB Math AI HL)

Key ideas

• In y = a e^−kt, a is the value at t = 0 and k > 0 is the decay rate.
• Linearising: if a plot of ln y against t is close to a straight line, an exponential model fits.
• From the regression of ln y on t: k = −(gradient) and a = e^intercept.
• Half-life: solve a e^−kt = a ÷ 2 to get t = ln 2 ÷ k.
• Cooling to room temperature: model the difference above room temperature, T − T_room, not T itself, because T does not decay to 0.
• The GDC's exponential regression gives the same kind of model directly; say which method you used.

Worked example

A cup of tea cools in a room at 20°C. The excess temperature y = T − 20 (°C) is recorded: t = 0, 10, 20, 30 minutes gives y = 60, 36.4, 22.1, 13.4. Fit a model y = a e^−kt using ln y, find the time for the excess temperature to halve, and estimate T after 40 minutes.

1. ln y = 4.094, 3.595, 3.096, 2.595 at t = 0, 10, 20, 30.
2. Regression of ln y on t: gradient ≈ −0.0500 and intercept ≈ 4.094.
3. So k ≈ 0.0500 and a = e^4.094 ≈ 60.0: y = 60.0 e^−0.0500t.
4. Half-life: ln 2 ÷ 0.0500 ≈ 13.9 minutes.
5. At t = 40: y ≈ 60.0 e^−2.00 ≈ 8.13, so T ≈ 20 + 8.13 ≈ 28.1°C.

Answer: y = 60.0 e^−0.0500t; the excess temperature halves every 13.9 minutes; T ≈ 28.1°C after 40 minutes.

Exam-style HL AI questions

• HL AI Number and Algebra questions by topic — worked examples and practice questions for every topic in this unit

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Regression on your calculator

See it on a graph

• Graph of y = e⁻ˣ
• Exponential regression
• Open the free Grapher

Maths IA ideas that use exponential models for real-world decay

• Radioactive decay with dice: does chance produce an exponential curve?
• How long would it take to guess your password? Counting and logarithms
• Exponential or logistic? Modelling my country's population
• How quickly does a swing die down? Exponential or linear decay?
• All HL AI IA ideas

Related in Number and Algebra

• Matrices — multiplication, inverse, and determinant
• Systems of linear equations and Gaussian elimination
• Complex numbers and Argand diagrams

Revise Exponential models for real-world decay for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S22 · Problem S86 · Problem S93 · Problem S28. All 22 →