Maths meets the real world
Mathematical modelling turns a messy real situation into maths, solves it, and turns the answer back into real-world meaning. Engineers, economists, epidemiologists, climate scientists and city planners all model.
The modelling cycle
| Stage | Questions |
|---|---|
| 1. Identify the problem | What do we want to know or decide? |
| 2. Make assumptions | What can we simplify or ignore? |
| 3. Define variables | What quantities matter? Units? |
| 4. Build the model | Which equations, functions or simulations fit? |
| 5. Solve | Calculate, graph or simulate. |
| 6. Interpret | What does the answer mean in context? |
| 7. Validation | Does it match real data or common sense? |
| 8. Refinement | Improve the model and repeat. |
Example 1: planning a school fundraiser
Problem: How many pizzas to order for a lunch sale?
- Assumptions: 60% of 800 students buy; each buys 2 slices; 8 slices per pizza.
- Model: pizzas = 0.6 × 800 × 2 ÷ 8 = 120 pizzas
- Sensitivity: if only 50% buy, 100 pizzas; if 70%, 140. The answer depends strongly on the participation assumption.
- Validation: compare with last year’s sales.
Example 2: phone battery
A phone loses about 8% of its full charge per hour of video. Starting at 100%, when does it reach 20%?
- Linear model: B(t) = 100 − 8t → 20 = 100 − 8t → t = 10 h
- Real batteries don’t drain perfectly linearly, so the model is an approximation.
Example 3: spread of a rumour
At first, a rumour spreads quickly, then slows as most people have heard it. A logistic (S-shaped) model fits better than an exponential one:
P(t) = L / (1 + ae^(−kt))
where L is the total population. Exponential models work early on but predict impossible numbers later.
Example 4: optimizing a design
A company wants a 355 mL can using the least aluminium. Model surface area as a function of radius, use calculus, and find r ≈ 3.8 cm and h ≈ 7.7 cm. Real cans are taller (about 6.6 cm diameter, 12.2 cm height) because of grip, stacking, shipping and tradition. The model guides decisions but isn’t the whole story.
Choosing model types
| Situation | Model |
|---|---|
| constant rate | linear |
| constant percent change | exponential |
| peak or optimum | quadratic or calculus |
| cycles | sinusoidal |
| limited growth | logistic |
| uncertainty | probability or simulation |
Communicating models
State assumptions, show the model, give answers with units, discuss limitations and how sensitive results are.
Worked examples
Example 1: A car uses 7 L/100 km. Model fuel F for distance d? F = 0.07d.
Example 2: Why might a linear model fail for population over 100 years? Growth tends to be percent-based (exponential) or limited (logistic).
Example 3: What’s one assumption in "the ball’s height is h = −4.9t² + 15t"? No air resistance (or it’s thrown from ground level).
Common mistakes
- Hiding assumptions.
- Trusting a model outside its valid range.
- Skipping validation.
Try it yourself
- Name the stage: comparing predictions to real measurements.
- A model predicts 140% attendance. What should you do?
- Which model type for daily high temperatures over a year?
- Give one assumption in a model of how long a candle burns.
Answers: 1) validation 2) Refine it; something’s wrong (an assumption or range). 3) sinusoidal 4) e.g. it burns at a constant rate.
Quick recap
- Mathematical modelling cycles through assumptions, variables, solving, validation and refinement.
- Choose model types that fit the pattern.
- Check sensitivity and state limitations.