A Resource for Free-standing Mathematics QualificationsGas Guzzlers
The table gives the number of cars with engine sizes
of 2 litres or more that were licensed in Great Britain
at the end of each year from 1994 to 2007.
Thesevalues are plotted on the graph below.
What type(s) of function do you think could be used to model these data?
The values from the Data Sheet are repeated in the table below.
In this table t represents the number of years since the end of 1994 (so that 0 represents the end of 1994, 1 represents the end of 1995 and so on) and N represents the number of thousands of cars with engine sizes of 2 litres or more that were registered.
These data can be modelled by an exponential function of the form .
Taking natural logarithms and using the laws
of logarithms:
Compare this with the linear equation
This suggests that a graph of ln N (on the y axis)
againstt(on the x axis) should give a straight line.
Assuming this is so, then the gradient will give
the value of k and the intercept on the y axis will
give the value of ln N0
- Complete the ln N column in the table.
- Draw a graph of ln N against t.
- Draw the line of best fit.
- Use the line of best fit to complete the following:
The gradient givesk = …………………………….
The intercept on the ln N axis gives:ln N0 = …………………………….
N0 = …………………………….
The exponential model isN= …………………………….
- How well does your model fit the data?
- What does the model predict for the years 2008 and 2009?
The actual values for 2008 and 2009 were 3731 thousand and 3768 thousand respectively. Comment on the accuracy of your predictions.
- Draw a new graph for the years 2000 to 2009.
Find a new function to model these data.
UnitAdvanced Level, Working with graphical and algebraic techniques
Notes on Activity
The data used in this activity can be introduced using the first two slides in the accompanying Powerpoint presentation. These slides and page 1 of this document give the number of cars with engines of 2 litres and above that werelicensed in Great Britainat the end of each year from 1994 to 2007. There are a variety of ways in which you could use these data depending on whether you want to use a log graph to find an exponential model (the intended focus of this activity) or whether you prefer to use the data more generally to find and discuss a wider selection of models.
Page 1 gives the data and graph,then asks students to suggest possible types of functions that could be used to model the data. As well as exponential functions, students might also suggest linear, quadratic or other polynomial functions. If you prefer to use the data for a general discussion about models, students couldbe asked to use a graphic calculator or spreadsheet to find one or more models and compare them. Advise students to use t to represent the number of years after 1994 - thisgives more user-friendly coefficients.
The following functions were obtained from a graphic calculator:
(r = 0.991), , ,
, (r = 0.999)
Students could compare the accuracy of the models they find using graphs or percentage errors. All of the above models agree reasonably well with the data.
If you prefer to concentrate on the exponential model, then Page2 provides a worksheet that shows students how to use natural logarithms to find such a model from a linear graph. This method is also explained on slide 3 of the Powerpoint presentation. The later slides give the solution – these can be used as a demonstration or an on-going check of students’ work.
Answers to Worksheet
The completed table and log graph are shown below with the line of best fit and its equation.
The gradient of the line of best fit givesk = 0.0704
The intercept givesln N0 = 7.318 and so = 1507
The exponential model is
Comparison of data and values predicted by the model
The graph below shows how well the model fits the data.
This can also be seen from the table which gives the percentage error for each of the data values.
Predictions for 2008 and 2009
Substituting t = 14 in gives thousand
Substituting t = 15 in gives thousand
These predictions are much greater than the actual values 3731 thousand and 3768 thousand.
Possible reasons for this are that the level of demand for cars with large engines did not increase as much as predicted because they consume a lot of petrol and are very costly to run, because global warming measures taken by the government may discourage their purchase etc. In general, this shows that extrapolating beyond the data used to find the model may give a poor prediction and that it might be advisable to find a new model. The final request on the worksheet is to use the data for 2000 – 2009 to find a new model. The graph for 2000 – 2009 and possible models are given below.
Graph for 2000 to 2009 data
Possible models
The graphs below show models suggested by Excel. The cubic function is the best of these.
The following functions, which are very similar to those above, were obtained from a graphic calculator:
(r = 0.986), , ,
(r = 0.978)
The Nuffield Foundation
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