Mathematical Modeling and Problem Solving
Core concepts
Modelling translates a real situation into mathematics, solves it, and interprets the result in context. You must define variables clearly, state assumptions, choose an appropriate function family (linear, quadratic, exponential, trigonometric), and validate the model against the data or situation.
Common tasks include fitting a curve to data, interpreting parameters (initial value, growth rate, amplitude, period), and using calculus or algebra to optimise. Interpretation is as important as computation: an answer of 3.7 people should be reported as 3 or 4 depending on context, and negative time or negative length must be rejected.
Traps to watch
- Losing track of units, for example mixing metres and centimetres in one formula.
- Reporting an unrounded calculator value when the context demands a sensible rounding.
- Ignoring the physical domain, such as accepting a negative time solution.
- Choosing an unsuitable model family, e.g. a linear model for clearly exponential growth.
Rule of thumb
Write down what each variable represents and its units before solving. After solving, ask whether the answer is physically sensible and state it in context.
Worked example
A population grows from 500 to 800 in 4 years. Assuming exponential growth, 500·b⁴ = 800 gives b⁴ = 1.6, so b ≈ 1.125, a growth rate of about 12.5% per year. The model predicts 500(1.125)^t individuals after t years, and predictions far beyond the data should be treated cautiously.