9 A biologist is investigating the growth of bacteria in a piece of bread.
He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.
- Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line.
Give, in terms of \(A\) and \(k\),
- the gradient of the line
- the intercept on the vertical axis.
The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 . - Find the values of \(A\) and \(k\).
- Use the model to predict the number of bacteria after 24 hours.
- Give a reason why the model may not be appropriate for large values of \(t\).