What is a modelling assumption?

A simplification, such as air resistance being ignored or a rate being constant.

What are the main steps of modelling?

Translate the situation, use the model with inputs, interpret outputs, then refine and evaluate.

Typical assumptions for h=1.5+12t-5t^2?

Ball is a particle, air resistance negligible, gravity constant.

£40 is a fixed charge and £25 is the cost per day.

Interpret P=200\times1.5^t.

200 is the initial population and 1.5 is the growth factor per hour.

Why can a model fail for large inputs?

Its assumptions, such as unlimited growth, stop being true, or it gives impossible outputs.

What shows a model is used outside its valid domain?

Impossible outputs, such as a negative number of visitors or a height below ground.

How can you refine a model?

Change the assumptions, adjust the parameters or choose a different type of function.

What does it mean to evaluate a model?

To judge whether it is appropriate, using its assumptions and how well it fits data.

Why compare model outputs with real data?

It shows whether the assumptions are reasonable and where the model needs changing.

Mathematical modellingAQA A-Level Maths: Flashcards

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What is a mathematical model?

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What is a mathematical model?: A mathematical description of a real situation, made using simplifying assumptions.
What is a modelling assumption?: A simplification, such as air resistance being ignored or a rate being constant.
What are the main steps of modelling?: Translate the situation, use the model with inputs, interpret outputs, then refine and evaluate.
Typical assumptions for $h=1.5+12t-5t^2$ ?: Ball is a particle, air resistance negligible, gravity constant.
Interpret $C=40+25d$ .: £40 is a fixed charge and £25 is the cost per day.
Interpret $P=200\times1.5^t$ .: $200$ is the initial population and $1.5$ is the growth factor per hour.
Why can a model fail for large inputs?: Its assumptions, such as unlimited growth, stop being true, or it gives impossible outputs.
What shows a model is used outside its valid domain?: Impossible outputs, such as a negative number of visitors or a height below ground.
How can you refine a model?: Change the assumptions, adjust the parameters or choose a different type of function.
What does it mean to evaluate a model?: To judge whether it is appropriate, using its assumptions and how well it fits data.
Why compare model outputs with real data?: It shows whether the assumptions are reasonable and where the model needs changing.
Maximum of $h=1.5+12t-5t^2$ ?: $8.7$ m at $t=1.2$ s.

Exam questions on Mathematical modelling

A ball is thrown upwards from a height of 1.5 m. Its height $h$ metres after $t$ seconds is modelled by $h=1.5+12t-5t^2$ for $t\ge0$ .
The model is only valid for part of the ball's flight. Find the range of values of $t$ for which it is valid, giving your reasoning.2 marks
A population of bacteria is modelled by $P=200\times1.5^t$ , where $P$ is the number of bacteria and $t$ is the time in hours after the start of an experiment.
The model predicts about $665\,000$ bacteria after 20 hours, but in the experiment the population levels off at about 5000. Explain why the model fails, and suggest a refinement.2 marks
A van hire company's cost model is $C=40+25d$ , where $C$ is the cost in pounds and $d$ is the number of days of hire.
Interpret the numbers $40$ and $25$ in the model, and state one assumption it makes.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).