L46 — Mathematical Modelling in Context
Key Terms
- Mathematical model
- An equation, graph, or formula that represents a real-world situation so it can be analysed mathematically.
- Variable
- A quantity that can change; must be clearly defined (e.g. "let h = hours worked") before writing a model.
- Assumption
- A simplification of reality that makes the maths tractable — always state assumptions so the model's limitations are clear.
- Formulate
- The stage of the modelling cycle where variables are defined and equations are written based on the real situation.
- Interpret
- Translating a mathematical answer back into the language and units of the real-world context.
- Validate
- Checking whether the model's predictions are reasonable; identifying where the model breaks down or needs refinement.
The Mathematical Modelling Cycle
| Stage | What you do | Example |
|---|---|---|
| Formulate | Define variables, state assumptions, write equations | Let x = hours worked, C = 15x + 20 |
| Solve | Apply mathematical techniques to find answers | Solve C = 95: 15x + 20 = 95 ⇒ x = 5 |
| Interpret | Translate the answer back into context | 5 hours of work to earn $95 |
| Evaluate | Check reasonableness; identify limitations of the model | Does the model hold for very large or very small values? |
Common model types
| Situation | Model type | General form |
|---|---|---|
| Constant rate of change | Linear | y = mx + c |
| Projectiles, area, profit | Quadratic | y = ax² + bx + c |
| Population, compound interest | Exponential | y = A × rn |
| Proportional relationships | Inverse variation | y = k/x |
Worked Example — Plumber's Call-Out Fee
Situation: A plumber charges a $60 call-out fee plus $45 per hour of work.
Step 1 — Define variables and formulate.
Let h = number of hours worked. Let C = total cost ($).
Model: C = 45h + 60
Step 2 — Solve.
For a 3-hour job: C = 45(3) + 60 = 135 + 60 = $195.
Step 3 — Interpret.
A 3-hour job costs $195. The model is linear with gradient $45/hr and y-intercept $60.
Step 4 — Evaluate.
Assumption: constant rate; no travel time; whole hours. For very long jobs the rate might change — this model has limitations.
Why Do We Model?
Real problems are messy. A shopping centre wants to know how much car park space to build. A sports scientist wants to predict how far a ball will travel. A business owner wants to know how many units to produce to maximise profit.
Mathematics gives us a way to simplify these situations into equations we can solve — then we can use the answer to make decisions in the real world. This is mathematical modelling.
Choosing the Right Model
The first question in modelling is: what type of relationship is this?
- If the output changes by a constant amount for each unit increase in input → linear model.
- If the output reaches a maximum or minimum, or the rate itself is changing → quadratic model.
- If the output grows or decays by a constant factor each time → exponential model.
Look at the data or description for clues. A "per hour" rate suggests linear. "Squared" or "area" often suggests quadratic. "Doubles every year" is exponential.
Worked Example 2 — Ball Thrown Upward
A ball is thrown upward from a cliff 20 m high. Its height h (metres) after t seconds is modelled by h = −5t² + 20t + 20.
Maximum height: vertex at t = −20/(2 × −5) = 2 s. h(2) = −5(4) + 40 + 20 = 40 m.
When does it hit the ground? Set h = 0: −5t² + 20t + 20 = 0 ⇒ t² − 4t − 4 = 0 ⇒ t = (4 + √32)/2 ≈ 4.83 s (taking positive root).
Assumption: no air resistance; ball is a point mass.
Worked Example 3 — Bacteria Growth
A bacterial colony starts with 500 cells and doubles every 3 hours. Model the population P after n hours.
Every 3 hours: multiply by 2. After n hours: n/3 doubling periods.
P = 500 × 2n/3.
After 12 hours: P = 500 × 24 = 500 × 16 = 8 000 cells.
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Formulating a linear model. Fluency
A taxi company charges a $3.50 flag fall (starting fee) plus $2.20 per kilometre.
- (a) Define variables and write a cost model C(k) for a trip of k kilometres.
- (b) Find the cost of a 15 km trip.
- (c) If a customer pays $30.10, how far did they travel?
- (d) State one assumption in your model.
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Reading a model from a table. Fluency
A mobile phone plan charges per GB of data used:
Data used (GB) 0 1 2 3 5 Monthly cost ($) 25 35 45 55 75 - (a) What type of model does this data suggest?
- (b) Write a rule C(g) for the cost when g GB are used.
- (c) Find the cost for 7 GB.
- (d) What does the y-intercept represent in context?
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Linear model — two rates. Fluency
Company A charges $50 call-out + $30/hr. Company B charges $20 call-out + $40/hr.
- (a) Write cost models A(h) and B(h) for h hours of work.
- (b) Find the number of hours at which both companies cost the same.
- (c) For a 4-hour job, which company is cheaper?
- (d) For a 1-hour job, which is cheaper?
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Quadratic model — projectile. Fluency
A ball is kicked from ground level. Its height h metres at time t seconds is h = −4t² + 16t.
- (a) Find the height at t = 1, t = 2, t = 3.
- (b) When does the ball reach its maximum height? What is the maximum height?
- (c) When does the ball land?
- (d) Is this a realistic model? What assumption does it make?
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Exponential model — savings. Understanding
Mia invests $2 000 in an account earning 5% p.a. interest, compounding annually. The balance after n years is B = 2000 × 1.05n.
- (a) Find the balance after 3 years and after 10 years (to the nearest cent).
- (b) After how many complete years will the balance first exceed $3 000?
- (c) What does the "1.05" represent in the model?
- (d) What assumption does this model make about the interest rate?
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Model validation. Understanding
A student models the distance d (km) a car travels on a full tank as d = 12f, where f is litres of fuel.
- (a) What type of model is this? What does the gradient represent?
- (b) If a tank holds 60 L, what distance does the model predict?
- (c) The car actually travels 680 km on a full tank. Is the model accurate? Calculate the percentage error.
- (d) Give one reason the actual distance might differ from the model.
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Building a model from a description. Understanding
A swimming pool is being filled. At the start it has 2 000 litres. Water is added at 300 litres per minute.
- (a) Write a model V(t) for the volume after t minutes.
- (b) The pool holds 20 000 L. How long will it take to fill?
- (c) Sketch a graph of V(t), labelling the axes, gradient, and y-intercept.
- (d) At t = 30 min, the tap is turned off. Modify your model to show V is constant after t = 30.
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Piecewise model. Understanding
An electricity tariff charges 28 cents/kWh for the first 500 kWh used per quarter, and 35 cents/kWh for every kWh above 500.
- (a) Write a piecewise cost model C(u) for usage of u kWh.
- (b) Find C(400) and C(700).
- (c) What is the cost of exactly 500 kWh?
- (d) A household pays $238. How many kWh did they use?
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Choosing and justifying a model. Problem Solving
A coffee shop records daily customers over two weeks:
Day 1 2 3 4 5 6 7 Customers 42 47 51 54 56 57 58 - (a) Is this data better modelled by a linear or non-linear model? Explain your reasoning.
- (b) First differences: calculate the daily increase. What pattern do you notice?
- (c) A linear model through day 1 and day 7 gives C ≈ 2.67d + 39.3. Predict customers on day 14 and day 30. Are these predictions reasonable?
- (d) Suggest a limitation of the linear model for this data.
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Full modelling problem. Problem Solving
A farmer wants to fence three equal adjacent pens using 120 m of fencing. Each pen shares a wall with its neighbour. Let x be the width and y be the length of each pen.
- (a) Show that the total fencing used is 4x + 2y = 120, and hence express y in terms of x.
- (b) Write a function A(x) for the total area of the three pens.
- (c) Find the value of x that maximises A, and state the maximum total area.
- (d) State any constraints on x (what values are physically valid?)