6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula.
\(P = 100 \mathrm { e } ^ { t }\)
- Use this model to answer the following.
- Find the value of \(P\) when \(t = 4\).
- Find the value of \(t\) when the population is 9000 .
- It is suspected that a more appropriate model would be the following formula.
\(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
- Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line.
Some observations of \(t\) and \(P\) gave the following results.
| \(t\) | 1 | 2 | 3 | 4 | 5 |
| \(P\) | 100 | 500 | 1800 | 7000 | 19000 |
| \(\log _ { 10 } P\) | 2.00 | 2.70 | 3.26 | 3.85 | 4.28 |
- On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
- Hence estimate the values of \(k\) and \(a\).