Mathematical modellingAQA A-Level Maths: Revision notes

A mathematical model translates a real situation into mathematics so that it can be explored. To build one:

1. Identify the quantities involved and the question to answer.
2. Make simplifying assumptions, which are statements about what is ignored or held constant (for example "air resistance is negligible" or "the rate is constant").
3. Choose variables and a type of relationship (linear, quadratic, exponential, and so on).
4. Write the model as an equation or inequality. Example: a ball thrown upwards from 1.5 m at 12 m s$^{-1}$ is modelled by $h=1.5+12t-5t^2$. This assumes the ball is a particle, air resistance is negligible and gravity is constant.

Once a model exists you put in inputs and obtain outputs, and use them to explore the situation. For $h=1.5+12t-5t^2$:

• $t=0$ gives the starting height $1.5$ m;
• completing the square, $h=-5(t-1.2)^2+8.7$, so the maximum height is $8.7$ m after $1.2$ s;
• solving $h=0$ gives the landing time $t=2.52$ s. For $C=40+25d$ (van hire), $d=4$ gives $C=\text{£}140$. The model can also be explored backwards: £240 allows $d=8$ days. Always give outputs with units and appropriate accuracy.

Numbers in a model have meaning in context.

• In $C=40+25d$, the constant £40 is a fixed charge and the coefficient £25 is the cost per day.
• In $P=200\times1.5^t$, $200$ is the initial population and $1.5$ is the hourly growth factor (a 50% increase each hour).
• In $N=-2t^2+60t+100$ for website visitors, $N(0)=100$ is the number of visitors on launch day, and the maximum $N=550$ occurs on day 15. Interpretation means saying what a result means for the situation, not just stating it. Check that outputs are possible: a negative number of visitors or a height below ground shows the model has been used outside its valid domain.

Compare outputs with real data. If they disagree, refine the model by changing the assumptions, parameters or type of function.

• Exponential growth $P=200\times1.5^t$ predicts about $665\,000$ bacteria after 20 hours with no limit; a better model has an upper limit because food and space run out.
• A van hire model $C=40+25d$ that ignores discounts can be refined: if days after the 7th cost £15, then $C=110+15d$ for $d>7$.
• For website visitors, the quadratic model gives $N(40)=-700$, which is impossible, so it is restricted to $0\le t\le30$ and replaced by a model that levels off at a positive value for later days. To evaluate whether a model is appropriate, consider whether the assumptions are reasonable, whether the outputs agree with observed data, and for which inputs it remains valid.