Estimation & Modelling (AQA Level 3 Mathematical Studies (Core Maths): Paper 1: Data, Finance, Estimation & Modelling): Flashcards
Exam code: 1350
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- FrontThe Modelling Cycle
The modelling cycle has four stages. Complete the two words that are missing.
Represent the scenario mathematically, making assumptions to simplify it.
Use appropriate mathematical
to carry out the calculations.Interpret the results in the
of the original problem.Evaluate the methods and solutions by comparing them with real life.
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Cards in this collection (28)
The modelling cycle has four stages. Complete the two words that are missing.
Represent the scenario mathematically, making assumptions to simplify it.
Use appropriate mathematical
to carry out the calculations.Interpret the results in the
of the original problem.Evaluate the methods and solutions by comparing them with real life.
The completed stages are:
Represent the scenario mathematically, making assumptions to simplify it.
Use appropriate mathematical techniques to carry out the calculations.
Interpret the results in the context of the original problem.
Evaluate the methods and solutions by comparing them with real life.
The cycle then returns to the start, so the model can be refined and run through again.
Why does a mathematical model never include every real-life factor?
There are far too many factors and variables in real life to model them all precisely.
A model has to be practical and easy to use as well as accurate, so some factors are simplified and others ignored altogether.
Why must you understand how a problem works before choosing a technique?
The right technique depends on the underlying mechanism, and two problems that look alike can need quite different mathematics.
A finance problem needs you to know whether the interest is simple or compound, and whether it is applied before or after each payment.
A calculation says 41.67 containers of ice cream are needed. How many should be bought, and why?
42 containers, because only a whole number can be bought and 41 would not be enough.
Rounding in context follows what the situation allows, not the ordinary rounding rule.
Define an assumption in a mathematical model.
An assumption is a simplification you deliberately make about a real-life factor so that a situation can be modelled mathematically.
It is stated openly rather than left hidden, so that its effect on the answer can be judged later.
What should you check when evaluating a mathematical model?
Check whether the assumptions and simplifications were sensible, and what effect each one had on the answer.
Check too whether the mathematical techniques used were appropriate, and how close the solution came to real life.
What decides which factors a model keeps and which it ignores?
The purpose of the model decides it, because a factor that matters for one use may be irrelevant for another.
A model of a car covering 200 miles needs very different detail for a family going on holiday than for a Formula 1 team.
A model truck is a 1:76 scale replica of a real one. What scale factor connects their two volumes?
The volume scale factor is the cube of the linear scale factor, so it is
.The real trailer therefore holds
times the volume of the model trailer, not 76 times it.True or False?
Once a model produces a number, that number is the answer to the problem.
False.
The result still has to be read back into the original situation, which may add constraints the calculation ignored entirely.
A paint calculation can give the cheapest combination of tins arithmetically, until a "buy four, get the fifth half price" offer makes a different combination cheaper.
Why is the modelling process drawn as a cycle rather than a straight line?
Evaluating a model shows where its assumptions were weak, and that feeds straight back into representing the scenario again.
Each loop of the cycle produces a more accurate model than the one before it.
A van's volume divided by a box's volume gives 155.2 boxes. Why is that an over-estimate?
Dividing the volumes assumes the boxes fill the space perfectly, when in fact whole boxes leave gaps that nothing can fill.
Fitting boxes along each dimension separately gives 8 by 4 by 4, which is only 128 boxes.
What kinds of limit or condition might a modelled variable have?
A variable can have a maximum, such as a container's capacity or the hours a person may work, or a minimum such as the stock that must be kept.
Some variables also cannot be negative at all, since a population or a length has to be zero or more.
When estimating the number of trees in a wood, which two factors are essential and what would you assume?
The essential factors are the area of the wood and how densely the trees are packed into it.
You would assume the density is the same throughout, and approximate the wood by a simple shape such as a rectangle or a circle.
A model recommends buying a whole season's ice cream in one order. Why might that be poor advice?
The result ignores practical constraints: that much ice cream needs somewhere to be stored, and all of it has to be kept frozen.
The freezing alone would cost money in energy, which the volume calculation says nothing about.
A model gives 155.2 boxes and the answer is written down as 156. What is wrong with that?
Only a whole number of boxes fits, so the answer has to be rounded down to 155.
Rounding up would be claiming that a box which physically does not fit has been fitted into the van.
A construction project is being modelled. Give an example of an assumption, and something the model could then predict.
Assumptions could be made about the number of workers, their working hours and their rates of pay.
The model could then predict the total cost of the project and its estimated completion time.
Define Fermi estimation.
Fermi estimation is making a fast, rough estimate of a quantity that is difficult or impossible to measure directly.
How many leaves are on a typical tree, or how many bricks built the Great Wall of China, are the kinds of question it answers.
What is the first thing to do when starting a Fermi estimate?
Pin down the scope of the question so that it is well defined.
"How much pizza is eaten in the world?" becomes "how many tonnes of pizza are eaten globally each day?", which fixes both a unit and a time period.
True or False?
A Fermi estimate is a kind of mathematical model.
True.
A Fermi estimate identifies the factors that affect the answer, simplifies each one into a rough value, and combines them, which is exactly what building a mathematical model involves.
What makes a factor suitable for use in a Fermi estimate?
A usable factor is one that affects the answer and that you can put a rough number to.
A factor you cannot estimate at all is no use in the calculation, however important it may be.
Complete the conversions that Fermi estimates rely on.
There are 1000 kg in one
of mass, and 1000 m in one of distance.There are 1000 millilitres in one
of volume.The completed conversions are:
There are 1000 kg in one tonne of mass, and 1000 m in one kilometre of distance.
There are 1000 millilitres in one litre of volume.
Getting a conversion wrong shifts the answer by a factor of a thousand, which swamps every other estimate in the calculation.
Roughly what populations should you know for a Fermi estimate?
The world holds around 8 billion people and the UK around 67 million.
A city runs to hundreds of thousands or millions, a town to thousands or tens of thousands, and a village to hundreds.
True or False?
Two people doing the same Fermi estimate should arrive at the same answer.
False.
Different but equally sensible assumptions lead to different answers, and the estimates are deliberately rough.
What matters is that each assumption is reasonable and that the calculation follows correctly from it.
How do you check whether a Fermi estimate is reasonable?
Look at its order of magnitude and ask whether the answer ought to be in the thousands, the millions or the billions.
An answer of 8 billion kg of pizza a day would mean every person on Earth eating a kilogram of it daily, which is clearly wrong.
Roughly how tall is a person, and how large is a football field?
A person is about 1.6 to 1.8 m tall, and a house about 8 to 10 m.
A football field covers around 10 000 square metres, against roughly 1 square metre for a table.
About 8 billion people exist, half of them eat one 0.5 kg pizza a week. Estimate the mass eaten globally each day.
Half of 8 billion gives 4 billion pizzas a week, which is roughly 600 million a day.
At 0.5 kg each that is 300 million kg, or about 300 000 tonnes every day.
Roughly what volumes should you know for a Fermi estimate?
A cup holds about 250 ml, a bath about 150 litres, and an Olympic swimming pool about 2500 cubic metres.
A thousand litres make one cubic metre, which is the conversion those three all sit on.
How would you estimate how many swimming pools the water drunk by a city in one week would fill?
Estimate the city's population, the size of a cup and how many cups a person drinks a day, then multiply those together across seven days.
Divide that total by the volume of one pool, which you can either recall or work out from its dimensions.
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