Practice Maths

Mathematical Modelling

Key Ideas

Key Terms

Mathematical modelling
The process of using mathematics to represent a real-world situation in order to solve a problem or make predictions.
The modelling process

   1. Identify — understand the problem and determine what you need to find.
   2. Formulate — choose a mathematical structure (e.g. linear equation, formula, table) and state assumptions.
   3. Solve — apply mathematics to find a result.
   4. Evaluate — check whether the answer makes sense in context; consider limitations of your model.
   5. Communicate — report findings clearly in the context of the original problem.
linear model
Applies when the rate of change is constant: y = mx + c.
non-linear model
Needed when growth or change is not constant (e.g. compound interest, area formulas).
Assumptions
Simplify a model (e.g. “the car travels at a constant speed”). Always state them.
Limitations
The conditions under which the model may not apply or may be inaccurate.
Hot Tip A model is a simplification of reality. Always ask: “Is my answer reasonable?” and “What have I assumed that might not be true?”

Worked Example

Question: A plumber charges a $70 call-out fee plus $55 per hour. Model the total cost and find the cost of a 3.5-hour job.

1. Identify: Find the total cost for any job duration.

2. Formulate: Assumption: hourly rate is constant; no materials cost. Model: C = 70 + 55h, where h = hours and C = cost ($).

3. Solve: For h = 3.5: C = 70 + 55(3.5) = 70 + 192.50 = $262.50

4. Evaluate: The model is linear — suitable since the rate is constant. A 3.5-hour job costing about $260 seems reasonable.

5. Communicate: The plumber would charge $262.50 for a 3.5-hour job under these conditions. This assumes materials are not included.

What is Mathematical Modelling?

Mathematical modelling is the process of representing a real-world situation using mathematical language — equations, graphs, formulas, or diagrams — in order to understand it, make predictions, or solve a problem. Scientists model the spread of diseases, engineers model stress on a bridge, and economists model market trends. At Year 9 level, modelling problems ask you to connect real situations to the mathematics you know.

The Modelling Cycle

Mathematical modelling follows a cycle of four stages:

  • Formulate: Understand the real-world problem. Identify the variables (what changes?), state your assumptions (what are you simplifying?), and translate the situation into a mathematical form (equation, formula, table, etc.).
  • Solve: Use mathematical techniques to solve the model. This might involve solving an equation, calculating an area, using Pythagoras' theorem, or reading a graph.
  • Evaluate: Check whether your mathematical solution makes sense in the real world. Is it reasonable? Did your assumptions affect the answer? What are the limitations?
  • Communicate: State your answer clearly in the context of the original problem — with units, and in plain English.

Setting Up a Model

The formulate step is often the hardest. Look for key information: what quantity are you trying to find? What do you know? What formula or rule connects them?

Example: A farmer wants to fence a rectangular paddock with exactly 120 m of fencing. One side runs along a river (no fence needed). What dimensions maximise the area?

Variables: let width = x m. Then the opposite side is also x m, and the remaining fence goes across: length = (120 − 2x) m. Model: Area = x(120 − 2x) = 120x − 2x2. Now solve: this is a quadratic — maximum at x = 30, giving length 60 m and area = 30 × 60 = 1800 m2.

Assumptions and Limitations

Every model makes assumptions. Stating them is part of good mathematical communication. For the fencing example, we assumed the river bank is straight and the fencing is used in full. In reality, the ground might be uneven, there may be gates, and fencing comes in fixed lengths. These factors are ignored in the model — and that's fine, as long as we acknowledge it.

A limitation is a restriction on how well the model represents reality. Always ask: "Does my answer make physical sense? Are there constraints I haven't considered?"

Checking Reasonableness

After solving, check that your answer is sensible. If you calculate that a person is 45 metres tall, something has gone wrong. Use estimation, substitute your answer back into the original formula, and verify units are correct. A quick sanity check before writing your final answer catches many errors.

Key tip: In modelling questions, marks are available for every stage of the cycle — not just the final answer. Write a clear "define variables" statement (e.g. "let x = the number of hours"), show all working for the solve step, and write a full sentence for your communicate step. Students who only write the answer miss significant marks.

Mastery Practice

  1. A mobile phone plan charges a $15 monthly fee plus 8 cents per minute of calls. Fluency

    1. Identify the variables and write an equation modelling the monthly cost C (in dollars) in terms of the number of minutes called, m.
    2. State one assumption your model makes.
    3. Calculate the cost for a month where 250 minutes are used.
    4. If a customer’s bill was $39, how many minutes did they use?
    5. Sketch a graph of C versus m for 0 ≤ m ≤ 400. Label intercepts and gradient.

  2. For each situation below, state whether a linear or non-linear model is more appropriate and give a reason. Fluency

    1. A car travels at a constant speed of 80 km/h. Model the distance as a function of time.
    2. A bacteria culture doubles every hour. Model the population over time.
    3. Water drains from a tank at a constant rate of 20 L/min. Model the volume remaining.
    4. The area of a square depends on its side length.
    5. A shop charges a flat rate of $12 per item, regardless of quantity.

  3. The height h (in metres) of a ball thrown upward is modelled by h = −5t² + 20t + 2, where t is time in seconds. Fluency

    1. What is the initial height of the ball (at t = 0)?
    2. Calculate h when t = 1 and t = 3.
    3. Is this a linear or non-linear model? Explain.
    4. What does the negative coefficient of t² tell you about the motion?
    5. State one assumption the model makes.

  4. A school tuck shop sells pies for $3.50 each. The daily fixed costs (electricity, staff) are $80. The shop sells an average of 60 pies per day. Understanding

    1. Write an equation for the daily revenue R in terms of the number of pies sold n.
    2. Write an equation for the daily profit P (revenue minus fixed costs) in terms of n.
    3. How many pies must be sold to break even (P = 0)?
    4. Using the average sales of 60 pies, calculate the expected daily profit.
    5. State two assumptions your model makes. State one limitation.

  5. A farmer wants to enclose a rectangular paddock along a straight river. He has 120 m of fencing and will use the river as one side (no fence needed on that side). Understanding

    1. Let the side perpendicular to the river have length x metres. Write expressions for the length of the side parallel to the river and the area A in terms of x.
    2. Show that A = x(120 − 2x) = 120x − 2x².
    3. Complete a table of values for A when x = 10, 20, 30, 40, 50, 60.
    4. From your table, estimate the value of x that gives maximum area.
    5. State one assumption of your model and one limitation.

  6. A car uses petrol at a rate of 8.5 L per 100 km. Petrol costs $1.85 per litre. Understanding

    1. Write a formula for the petrol cost C (in dollars) to travel d km.
    2. Calculate the cost of a 350 km trip.
    3. If a driver has a budget of $50 for petrol, what is the maximum distance they can travel?
    4. The car’s fuel efficiency improves to 7.2 L/100 km after a service. How much would the driver save on the 350 km trip?
    5. State one assumption and one real-world factor your model ignores.

  7. A student models the number of people N who have heard a rumour after d days as N = 3d + 5. Understanding

    1. How many people have heard the rumour on Day 0 (when it starts)?
    2. According to the model, when will 50 people have heard the rumour?
    3. After 10 days the student observes 85 people have heard it, not 35. Identify the shortcoming of the linear model in this context.
    4. Suggest what type of model (linear or non-linear) would better represent the spread of a rumour. Give a reason.
    5. State one assumption of the original linear model.

  8. A swimming pool holds 80 000 litres of water. It is drained at a constant rate of 2 500 litres per hour for cleaning. Problem Solving

    1. Write an equation for the volume V (in litres) remaining after t hours of draining.
    2. How many hours will it take to fully drain the pool? Show your working.
    3. The pool must also be refilled after cleaning. The refill rate is 3 200 L/hr. Write an equation for the volume during refilling, starting from t = 0 at the start of refilling.
    4. If draining starts at 8:00 am, at what time will the pool be full again?
    5. The pool maintenance company charges $45 per hour for draining and $38 per hour for refilling. Calculate the total cost of the clean.

  9. Two hire car companies have the following pricing:
    Company A: $45/day + $0.18/km    Company B: $30/day + $0.26/km Problem Solving

    1. Write cost equations for each company (cost C in dollars, distance d in km, for a 1-day hire).
    2. Find the distance at which both companies charge the same amount.
    3. For a trip of 150 km in one day, which company is cheaper and by how much?
    4. A traveller plans to drive 300 km in one day. Which company should they choose? By how much?
    5. Explain the role of the gradient and y-intercept in each model, in the context of the problem.

  10. A landscape gardener is designing a rectangular garden bed. The garden must have a perimeter of 28 m and the client wants to maximise the planted area. Problem Solving

    1. Let the width of the garden be w metres. Write the length l in terms of w using the perimeter constraint.
    2. Write the area A as a function of w only.
    3. Expand and simplify: A = w(14 − w).
    4. Use a table of values (w = 1, 2, …, 7) to find the width that maximises area.
    5. What shape does the maximum-area rectangle turn out to be? What does this tell you generally?
    6. State the maximum possible area and confirm it satisfies the perimeter condition.
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