13 The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years.
| Year | \(1986 / 87\) | \(1991 / 92\) | \(1996 / 97\) | \(1999 / 00\) | \(2000 / 01\) | \(2001 / 02\) |
| Percentage of the | | adult population | | visiting the cinema |
| 31 | 44 | 54 | 56 | 55 | 57 |
Source: Department of National Statistics, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
This growth may be modelled by an equation of the form
$$P = a t ^ { b } ,$$
where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.
- Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(\log _ { 10 } t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
\section*{Answer part (ii) of this question on the insert provided.}
- Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(\log _ { 10 } t\). Draw by eye a line of best fit for the data.
- Use your graph to find the equation for \(P\) in terms of \(t\).
- Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\) ), according to this model.