Mathematical modelling and problem solving in applied contexts
Mathematical modelling and problem solving in applied contexts
This domain tests the full modelling cycle: translating a real situation into mathematics, stating assumptions, solving, interpreting the result and checking whether the model is reasonable.
Core rules
- Define variables clearly and give units. State assumptions such as treating an object as a particle, ignoring air resistance or assuming constant acceleration.
- Draw a diagram and identify given and unknown quantities before choosing a method.
- Check dimensions and units at each stage; an equation must be dimensionally consistent.
- Solve exactly where possible, then round only at the end to a sensible precision.
- Interpret the mathematical answer in the original context, including whether it is physically possible.
- Test sensitivity: small changes in assumptions or data should not produce wildly different conclusions; if they do, discuss the limitation.
Worked example
A rectangular pen is to be made against a straight wall using 40 m of fencing for the other three sides. Let x be the width perpendicular to the wall and y the length parallel to it. Fencing gives 2x + y = 40, so y = 40 − 2x. Area A = xy = 40x − 2x². Then dA/dx = 40 − 4x = 0, giving x = 10 and y = 20. The maximum area is 200 m²; checking endpoints confirms a maximum.
Common traps
Leaving assumptions unstated; applying a formula outside its conditions; mixing units; over-rounding during working; optimising without checking endpoints or feasibility; and giving a bare number without interpreting it in context.