log(y) vs x: convert and interpret

A student conducts an experiment and records the following data for two variables, \(x\) and \(y\).

\(x\)	1	2	3	4	5	6
\(y\)	14	45	130	1100	1300	3400
\(\log_{10} y\)

The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = kb^x\)

1. Plot values of \(\log_{10} y\) against \(x\) on the grid below. [2 marks] \includegraphics{figure_10}
2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly. [1 mark]
3. By drawing an appropriate straight line, find the values of \(k\) and \(b\). [4 marks]

Theresa bought a house on 2 January 1970 for £8000. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are shown in the following table.

Year	1970	1980	1990	2000	2010
Valuation price	£8000	£19000	£36000	£82000	£205000

The valuation price of the house can be modelled by the equation $$V = pq^t$$ where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants.

1. Show that \(V = pq^t\) can be written as \(\log_{10} V = \log_{10} p + t \log_{10} q\) [2 marks]
2. The values in the table of \(\log_{10} V\) against \(t\) have been plotted and a line of best fit has been drawn on the graph below. \includegraphics{figure_8b} Using the given line of best fit, find estimates for the values of \(p\) and \(q\). Give your answers correct to three significant figures. [4 marks]
3. Determine the year in which Theresa's house will first be worth half a million pounds. [3 marks]
4. Explain whether your answer to part (c) is likely to be reliable. [2 marks]

A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass \(M\) kg of salt remaining after \(t\) minutes by \(M = ak^t\) where \(a\) and \(k\) are constants.

1. Show that the model for \(M\) can be rewritten in the form \(\log_{10} M = t\log_{10} k + \log_{10} a\). [1]

The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.

\(t\) minutes	8	13	21	35	50
\(M\) kg	0.4	0.3	0.2	0.1	0.05

The student uses this data and plots \(y = \log_{10} M\) against \(x = t\) using graph drawing software. The software gives \(y = -0.0214x - 0.2403\) for the equation of the line of best fit.

1. 1. Find the values of \(a\) and \(k\) that follow from the equation of the line. [2]
   2. Interpret the value of \(k\) in context. [1]
2. It is known that when \(t = 0\) the mass of salt in the bucket is 0.5 kg. Comment on the accuracy when the model is used to estimate the initial mass of the salt. [1]
3. Use the model to predict the value of \(t\) at which \(M = 0.01\) kg. [2]
4. Rewrite the model for \(M\) in the form \(M = ae^{-ht}\) where \(h\) is a constant to be determined. [2]

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows. These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.

1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]

A doctors' surgery starts a campaign to reduce missed appointments. The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.

Number of weeks after the start (\(x\))	1	2	3	4	5
Number of missed appointments (\(y\))	235	149	99	59	38

This data could be modelled by an equation of the form \(y = pq^x\) where \(p\) and \(q\) are constants.

1. Show that this relationship may be expressed in the form \(\log_{10} y = mx + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). [2]

The diagram below shows \(\log_{10} y\) plotted against \(x\), for the given data. \includegraphics{figure_5}

1. Estimate the values of \(p\) and \(q\). [3]
2. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable. [2]