3.11
consist of large numbers of equations and associated data, and are implemented on computers. They can predict (with varying success) things such as what will happen to the inflation rate if interest rates are raised.
Such models are gross simplifications because there are too many variables contributing to the condition of a national economy and all factors can never be included.
Scientists also use models, for example in predicting population growth. Such a model, for example, to predict fish stocks in fishing areas, can be invaluable as it may be used to control quotas on fish catches to ensure that fishing does not reduce stocks to
unsustainable levels.
In both of these cases, the model has been produced as a result of a problem-solving exercise. The actual development of a model to represent a process is beyond the multiple-choice questions in the lower-level thinking skills examinations and will be dealt with in Chapter 5.2. Multiple-choice questions on choosing and using models test some of the basic skills involved in modelling and the extraction of data from mathematical models.
In the following activity you are asked to use different models to compare calculations.
This example is close to a real-life situation.
Activity
mathematics. What effect would these changes have on someone earning $50,000 a year?
As an exercise, consider other possible tax structures which might give similar results.
Plotting graphs of tax paid against earnings gives a clearer representation of how the various models of taxation work. The graph below shows the tax paid under the current system of tax in the example above.
0 10,000 20,000 30,000 40,000 50,000 0
2000 4000 6000 8000 10,000 12,000
Annual earnings
Tax due
Add lines for the proposed new system and for any other model you may think of.
The next activity, below, requires you to go some way towards developing a mathematical model of a new situation in order to solve the question.
Commentary
The model of tax used here is quite simple, consisting of a fixed amount of income on which no tax is paid and a single standard rate on earnings above this amount. Currently, those earning $26,000 pay (26,000−2000) × 0.2 dollars, or $4800. If they are to pay the same under the new regime, they will pay tax on $16,000, and the total will be the same as before, i.e. $4800.
The new tax rate will be 30¢ on each dollar earned over $10,000 ($16,000 × 0.3 = $4800).
This could be done by algebra, but the process is no simpler than that given above, which requires no more than elementary school
The government is determined to reduce the tax burden on lower-paid people and intends to bring in a new system, which will mean that the threshold for paying tax will rise to $10,000. They intend that all those below the average earnings of $26,000 will pay less tax, and all those earning more than this will pay more tax. What will be the tax rate on earnings over $10,000?
Activity
My company regularly uses a taxi service to take staff to the airport. If there are several passengers needing to travel from our town at similar times, they combine this into a single journey. They divide the total cost by the number of passengers and invoice each passenger separately. The distance is always the same and the time only varies by a small amount, but I do not know how they work out the charge for the journey.
There are a number of different charging structures they could use. All taxis charge a fixed price per kilometre and per minute of journey time. In addition they may charge a fixed hire fee and an additional charge depending on the number of passengers carried.
What is the charging structure used by this taxi company? What limitations are there to the conclusions we can derive?
Number of passengers 1 2 3 4 5
Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38
Commentary
If we just look at the data as it stands the pattern is not clear, other than that the price per passenger drops with the number of passengers. Since we are looking at the charge made by the taxi company, it is preferable to look at the total cost of the taxi in each case.
This may be carried out by multiplying the cost per passenger by the number of passengers, as shown in the table below.
The pattern now becomes much clearer.
Allowing for some small variations (it was stated that there was a small variation in journey time), the first two values are the same and they then increase by $4 per passenger. We can, therefore, conclude that the taxi company hire fee includes one or two passengers, then there is an extra charge of $4 per additional passenger.
The $40 ‘basic’ fee covers the hire charge, the distance charge and an average time charge. We have no information which will enable us to separate these three items. A model can only be as good as the data on which it is based.
Number of passengers 1 2 3 4 5
Charge per journey per passenger $40.00 $19.98 $14.68 $12.03 $10.38 Total charge per journey $40.00 $39.96 $44.04 $48.12 $51.90
A graph can be a very useful tool for analysing data such as in the table below, and can also help in developing models. Try graphing the data, for both the cost per passenger and the total cost per journey. Does this help in clarifying the charging structure?
The activity above introduced the idea that models usually are approximations to the real world. The model used did not allow for variations in the time of the journey. This is why the word ‘model’ is used. Almost all models are approximate – the model car does not usually have an internal combustion engine. Economic models cannot take into account factors such as the weather.
Many people use models in their everyday lives without even realising it. An efficient shopkeeper will, for example, have a set of rules that tells her how much ice cream to order so she has plenty in the summer months and less stock in the winter.
• We have learned how a mathematical or graphical model may be used to approximate real-life processes.
• We have seen how models can be used to simulate changes in cost structure and their effects.
• We have used real data to calculate the constants used in a mathematical model, for example the starting rate and charge per mile for a taxi fare.
• A graph of any sort is a model from which it is possible to get a picture of how variations can occur.
Summary
2 Finn walks to school, a distance of 1.5 km which takes him 15 minutes. His older sister, Alice, cycles to school on the same route at an average speed of 18 km/h.
She leaves home 5 minutes later than Finn. Does she overtake him on the way and, if so, where? At what time would she have to leave to arrive at school at exactly the same time as Finn?
3 A shop normally sells breakfast cereal for
$1.20 a packet. It is currently running a promotion, so if you buy two packets, you get a third free.
Tabulate and graph the price per packet for numbers of packets bought from 1 to 10. How would other special offers (e.g.
‘Buy one, get one half price’) affect the shape of this graph? If you were working backwards from the graph, how could you determine which offer is currently being used?
Answers and comments are on pages 322–3.
1 A novelty marketing company is selling an unusual liquid clock. It consists of two tubes as shown. The right-hand tube fills up gradually so that it is full at the end of each complete hour, and then empties and starts again. The left-hand tube does exactly the same in 12 hours. The time shown on the clock is 9.15.
Draw what the clock looks like at 4.20.