12 Answer part (ii) of this question on the insert provided.
The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
| Years after proposal accepted \(( t )\) | 1 | 2 | 3 | 4 | 5 |
| Cost \(( \pounds y\) million \()\) | 250 | 300 | 360 | 440 | 530 |
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
- Show that \(y = a b ^ { t }\) may be written as
$$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
- On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
- Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
- According to this model, what was the estimated cost of the project when it was first accepted?
- Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.