10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).
| Time \(( t\) hours \()\) | 1 | 2 | 3 | 4 | 5 |
| Number of bacteria \(( N )\) | 120 | 170 | 250 | 370 | 530 |
- Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
- Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
- Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\). - Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).