8 Theresa bought a house on 2 January 1970 for \(\pounds 8000\).
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 | \(\pounds 8000\) | \(\pounds 19000\) | \(\pounds 36000\) | \(\pounds 82000\) | \(\pounds 205000\) |
The valuation price of the house can be modelled by the equation
$$V = p q ^ { t }$$
where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants.
8
- Show that \(V = p q ^ { t }\) can be written as \(\log _ { 10 } V = \log _ { 10 } p + t \log _ { 10 } q\)
8 - 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.
| \(t\) | 0 | 10 | 20 | 30 | 40 |
| \(\log _ { 10 } V\) | 3.90 | 4.28 | 4.56 | 4.91 | 5.31 |
\includegraphics[max width=\textwidth, alt={}]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-11_1927_1207_625_338}
Using the given line of best fit, find estimates for the values of \(p\) and \(q\).
Give your answers correct to three significant figures.
8 - Determine the year in which Theresa's house will first be worth half a million pounds.
8
- Explain whether your answer to part (c) is likely to be reliable.