Mathematical Modelling and Problem Solving in Social Science Contexts
This domain tests the translation of a statement in economic or social language into functions, equations and inequalities, and the interpretation of the result. Typical settings are costs, revenue and profit, price–demand functions, break-even analysis, tariffs, loans and populations. Marks are awarded for the setup, the calculus and the interpretation, not only for the number.
Procedure that earns marks
- Declare the variable with its units and domain: "x = units produced per month, x ≥ 0".
- Translate sentence by sentence: a fixed cost becomes a constant term, "per unit" becomes a coefficient of x, a price–demand relation becomes p(x), revenue is I(x) = x·p(x), and profit is B(x) = I(x) − C(x).
- Restrict the domain using the context (non-negativity, capacity, price ≥ 0) and say so explicitly.
- Optimise with the derivative, justify maximum or minimum with f'' or a sign table, and answer with units.
- Interpret every parameter: the intercept, the slope, the vertex.
Worked example
A firm has fixed costs of 400 € and a variable cost of 15 € per unit; the demand function is p = 60 − 0.5x euros. Then C(x) = 400 + 15x, I(x) = x(60 − 0.5x) = 60x − 0.5x², and B(x) = I(x) − C(x) = −0.5x² + 45x − 400 for 0 ≤ x ≤ 120. Setting B'(x) = −x + 45 = 0 gives x = 45, and B'' = −1 < 0 confirms a maximum. B(45) = −1012.5 + 2025 − 400 = 612.5 €. Break-even requires B(x) = 0: x² − 90x + 800 = 0, so x = 10 or x = 80 units.
Traps that cost marks
- Identifying revenue with profit, or writing revenue as x + p instead of x·p.
- Reporting the maximum profit without the production level that produces it, or the reverse.
- Optimising over all real numbers when the context forces x ≥ 0 or an upper capacity limit.
- Forgetting that break-even has two roots and that the profit is positive only between them.
- Leaving the answer as a bare number when the question says "interpret" or "justify".