Mathematical Modeling

This final chapter ties the course together. Modelling is the cycle of taking a real problem, representing it with an equation, function or graph you already know, solving it, and then interpreting whether the answer makes sense.

It draws on many earlier chapters at once.

Modelling questions are the most open in the paper: a real scenario you must translate into maths yourself. They reward students who can connect chapters, spotting that a worded problem is really a quadratic, or a rate, or a variation, and who interpret their answer rather than stopping at a number.

The hardest part of a modelling question is almost never the algebra, it is the first step of turning a paragraph of English into symbols, and rushing it is where marks are lost. Before you calculate anything, name your variables explicitly: write down what each letter stands for and its units, for example 'let V be the volume of water in litres and t the time in minutes'.

Then state the assumptions the model rests on, such as 'assume the tap drains at a steady rate'. These two moves do real work: clear variables stop you mixing up which number is which, and stated assumptions tell you when the model is allowed to apply.

Examiners reward this setup because it shows you understood the situation, not just the arithmetic, and it makes every later step easier to write and to check.

Modelling draws on the functions you already met in earlier chapters, and the pattern in the situation tells you which one to pick. If a quantity changes by the same amount every step, losing 8 litres each minute, earning RM50 each sale, the relationship is linear, of the form y = mx + c, where m is the steady rate and c is the starting value.

If a quantity is multiplied by a fixed factor each step, or the situation involves an area, a squared distance or a turning point, a quadratic or exponential shape fits better. A quick way to decide from a table of values is to look at the differences between consecutive outputs: a constant difference signals a linear model, while a constant ratio signals repeated multiplication.

Matching the observed pattern to a function you have already studied is usually faster and safer than trying to invent a relationship from scratch.

A model is only as good as the answer it gives back, so the final part of the cycle is checking that answer against reality before you commit to it. Three quick tests catch most errors.

First, units: if you were finding a time, is your answer in minutes or seconds as the question wanted, and does it read sensibly? Second, scale: is the number roughly the size you would expect, or has a slipped decimal made a tank drain in 6 seconds instead of 60?

Third, domain: does the answer respect the limits of the situation, time and volume cannot be negative, a number of people must be a whole number, and a model like V = 500 − 8t only makes sense while V stays at or above zero, here for 0 ≤ t ≤ 62.5. When a question tells you to 'comment' or 'interpret', these are exactly the judgements it wants: say what the answer means, whether it is realistic, and where the model stops being valid.

This example walks through the full modelling cycle on a draining-tank scenario: build the model, use it, and judge where it stops being valid.