12 Answer the whole of this question on the insert provided.
A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10 ^ { b t }\), where \(t\) is the time in years after 2000.
- Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
- The table gives the data for the population from 2001 to 2005.
| Year | 2001 | 2002 | 2003 | 2004 | 2005 |
| \(t\) | 1 | 2 | 3 | 4 | 5 |
| \(P\) | 7900 | 8800 | 10000 | 11300 | 12800 |
- Use your graph to find the equation for \(P\) in terms of \(t\).
- Predict the population in 2008 according to this model.