Mathematical Modeling and Problem Solving
What this domain tests
Mathematical Modeling and Problem Solving asks you to translate a real or described situation into mathematics, choose an appropriate model, solve it, and interpret the result in context with correct units. Expect linear, quadratic, exponential, and trigonometric models, optimization under constraints, rates, and multi-step problems where the setup is the hard part.
Core rules and concepts
- Model selection: constant rate → linear; constant second difference or a fixed maximum → quadratic; constant percent change or doubling time → exponential; repeating behaviour → trigonometric.
- Exponential model: A(t) = A₀bᵗ, where b = 1 + r for growth rate r, or b = 1 − r for decay; half-life uses b = (1/2)^(1/h).
- Quadratic vertex: for y = ax² + bx + c, the extremum occurs at x = −b/(2a); this locates maximum height or minimum cost.
- Optimization: write the objective as one variable using the constraint, then find the vertex or use calculus.
- Units and dimensional reasoning: track units through each step; a rate multiplied by time gives a quantity.
- Interpretation: an answer must be reported in the problem's context, with rounding appropriate to the situation.
Worked example
A farmer has 60 m of fencing to enclose a rectangular pen against an existing straight wall (no fence needed on the wall side). What dimensions maximize area?
Let the two sides perpendicular to the wall each be x, and the side parallel to the wall be y. Fencing covers 2x + y = 60, so y = 60 − 2x. Area A = xy = x(60 − 2x) = 60x − 2x². This is a downward parabola with a = −2, b = 60, so the maximum is at x = −b/(2a) = −60/(−4) = 15. Then y = 60 − 30 = 30. Maximum area = 15 × 30 = 450 m², with dimensions 15 m by 30 m.
Note that the maximum occurs when the two perpendicular sides together equal the parallel side - a common structural result for wall-adjacent pens.
Traps that cost marks
- Misreading which side is unfenced, or fencing all four sides.
- Using linear growth for a percentage-change situation.
- Forgetting that exponential decay uses a base below 1.
- Reporting a raw number without units or contextual interpretation.
- Rounding intermediate values and carrying the error into the final answer.