Mathematical Modeling Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mathematical Modeling.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The process of using mathematical structures — functions, equations, distributions — to represent, analyze, and predict real-world phenomena.

Building a mathematical version of reality to understand and predict.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: All models are wrong, but some are useful. Models simplify to illuminate.

Common stuck point: Model assumptions may not hold—results are only as good as the model.

Sense of Study hint: Write down: (1) what real-world quantity each variable represents, (2) what you are ignoring, and (3) when the model would break. This makes gaps visible.

Worked Examples

Example 1

easy
A taxi charges a base fare of \2.50 and \1.20 per kilometre. Write a mathematical model for the total fare F as a function of distance d (km), identify variables, and compute the fare for a 7 km ride.

Solution

  1. 1
    Identify variables: F = total fare (\), d$ = distance (km).
  2. 2
    Model: F(d) = 2.50 + 1.20d.
  3. 3
    For d = 7: F(7) = 2.50 + 1.20 \times 7 = 2.50 + 8.40 = 10.90.

Answer

F(d) = 2.50 + 1.20d,\quad F(7) = \$10.90
A mathematical model translates a real-world situation into equations. Identifying what changes (variables) and what is fixed (parameters) is the first step in building any model.

Example 2

medium
A population of bacteria doubles every hour. If the initial count is P_0 = 500, write a model for the population P(t) after t hours and find P(4).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A rectangle has perimeter P and length l. Express the width w as a function of P and l, then find w when P=30 and l=8.

Example 2

medium
A car travels at a constant speed v km/h. Write a model for distance d after t hours. If the car must reach a destination 240 km away in 3 hours, what speed is required?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

abstraction