A mathematical model turns a real-world situation β like a cooling cup of coffee, a population, or a tide β into an equation. Once you have the model, you can predict values, work out unknowns, and answer “what if” questions.
π
What you need to know
A model simplifies a real-world situation using a function
Pick the right type of function based on the situation (linear, quadratic, exponential, etc.)
Find unknown parameters by substituting given values
Initial value = function value when t = 0
Long-term value = limiting value as t β β (often an asymptote)
Determine a reasonable domain using real-life context (no negative time, etc.)
Always check units β if “P in thousands of dollars”, P = 3 means $3,000
The Modelling Workflow
Most modelling questions follow the same four-step process:
From Situation to Prediction
1
Identify type
Linear? Exponential? Trig? Match the model to the scenario.
2
Set up
Write the general form with unknown parameters (A, k, a, bβ¦).
3
Find parameters
Substitute the given data points to form equations. Solve.
4
Use the model
Substitute new values to predict, or solve for the input.
Common Model Types
Each function type matches a particular real-world pattern:
The clue is in the wording. “Compound interest” or “doubling every X” β exponential. “Reaches a maximum” β quadratic. “Repeats every X” β trigonometric. “Approaches a limit” β exponential decay with offset.
Decoding Question Words
Modelling questions use specific phrases that translate directly to maths. Memorise these:
Phrase β Mathematical Action
“Initially” / “to start with”
β
substitute t = 0
“After 5 minutes / years / days”
β
substitute t = 5
“Reaches a value of 30”
β
solve f(t) = 30
“Maximum / minimum”
β
turning point
“Long term” / “limiting value”
β
find asymptote (t β β)
“How long until⦔
β
solve for t
The Coffee Cooling Model
Let’s look at a classic model: a cup of coffee cooling in a room. The temperature follows:
T(t) = Aekt + 16, t β₯ 0
where T is the temperature in Β°C and t is time in minutes. The 16 represents the room temperature β the coffee can never cool below this.
Coffee Cooling: T(t) = 64ekt + 16
Worked Examples
β
Example 1 β Use a model to predict a value
A population is modelled by P(t) = 200e0.05t, where t is time in years. Find: (a) the initial population, (b) the population after 10 years.
Answer:
(a) “Initial” β substitute t = 0.P(0) = 200e0 = 200(1) = 200200(b) “After 10 years” β substitute t = 10.P(10) = 200e0.05(10) = 200e0.5β 200 Γ 1.6487 β 329.7β 330 (3 s.f.)Population must be a whole number, but in modelling we usually round at the end.
β
Example 2 β Find when a value is reached
Using the same model P(t) = 200e0.05t, find how long it takes for the population to reach 500.
Answer:
Step 1: set P(t) = 500.200e0.05t = 500Step 2: isolate the exponential.e0.05t = 500/200 = 2.5Step 3: take ln of both sides.0.05t = ln 2.5t = ln(2.5) / 0.05 β 0.9163 / 0.05 β 18.33t β 18.3 years (3 s.f.)
β
Example 3 β Find an unknown parameter from a single data point
A model is m(t) = at3 + 5, where m is mass in kg after t hours. After 2 hours the mass is 7 kg. Find a.
Answer:
Step 1: substitute the data point m(2) = 7.a(2)Β³ + 5 = 78a + 5 = 78a = 2a = 0.25So the model is m(t) = 0.25tΒ³ + 5.
β
Example 4 β Coffee cooling (full SME problem)
The temperature TΒ°C of a cup of coffee follows the model:
T(t) = Aekt + 16, t β₯ 0
where t is time in minutes. Initially the temperature is 80Β°C and 5 minutes later it is 40Β°C.
Use the data point T(5) = 40 with A = 64.64e5k + 16 = 4064e5k = 24e5k = 24/64 = 3/8Take ln of both sides.5k = ln(3/8)k = (1/5) ln(3/8)“Exact value” means leave it in log form β don’t decimalise.
(c) Find the time taken for the temperature to reach 30Β°C.
Set T(t) = 30 and solve for t.64ekt + 16 = 3064ekt = 14ekt = 14/64 = 7/32kt = ln(7/32)Divide by k = (1/5) ln(3/8).t = ln(7/32) / [(1/5) ln(3/8)]= 5 ln(7/32) / ln(3/8) β 7.7476β¦t β 7.75 minutes (3 s.f.)Keep k symbolic until the final step to avoid rounding errors.
β
Example 5 β Choose a suitable model
A scientist measures the height of a plant. After 1 day it’s 5 cm; after 2 days it’s 10 cm; after 3 days it’s 20 cm. Suggest a suitable model and explain.
Answer:
Step 1: look at how values change between consecutive days.5 β 10: doubles10 β 20: doublesStep 2: a constant ratio (Γ2 each day) means EXPONENTIAL growth.Suitable model: h(t) = A Γ 2t or h(t) = AektStep 3: find A using h(1) = 5.A Γ 2ΒΉ = 5 β A = 2.5h(t) = 2.5 Γ 2tCheck: h(2) = 2.5 Γ 4 = 10 β, h(3) = 2.5 Γ 8 = 20 β
π‘
Tips
Read the question twice. Identify what’s given (data points), what’s unknown (parameters), and what’s asked (predict / find when / find max).
“Initially” nearly always means substitute t = 0 β and the answer is usually one of the parameters.
Work symbolically as long as possible. Keep parameters like k in exact form until the final calculation.
Check the asymptote for exponential decay models: a “+ c” at the end is usually a real-life baseline (room temp, ground level, etc.).
Sketch the model if you’re stuck on the domain β it shows what’s reasonable.
Use your GDC for solving and substituting, especially in Paper 2.
Always include units in your final answer (minutes, Β°C, kg, etc.).
β
Common mistakes
Decimalising too early. If the question says “exact value”, leave answers in ln, Ο, or fraction form β don’t convert to a decimal.
Confusing input and output. “When does the temperature reach 30?” wants t, not T. Read carefully which variable is given.
Ignoring the domain. If t β₯ 0 is the realistic range, negative time isn’t a valid answer.
Wrong model type. If the data doubles each step β exponential. If it adds the same amount each step β linear. Don’t guess.
Forgetting the asymptote in cooling/decay models. Coffee cools to room temp, not to 0Β°C β that’s why there’s a “+ 16”.
Sub-in errors with brackets. Substituting t = 0 into Aekt gives A Γ 1 = A (not 0!). e0 = 1, not 0.
Missing units in the final answer. “7.75” alone isn’t enough β it’s “7.75 minutes”.
Final word: Modelling is just three steps in disguise β match the function to the situation, find the parameters from given data, then use the model. Read the question carefully, work symbolically, and always include units.
Need help with Functions?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
