- A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.
A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.
- Show that the scientist's model for \(p\) leads to the equation
$$p = a \mathrm { e } ^ { k t }$$
where \(a\) and \(k\) are constants.
The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish.
She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
- Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
- Hence show that the model for \(p\) can be rewritten as
$$p = a b ^ { t }$$
stating, to 3 significant figures, the value of the constant \(b\).
With reference to this model,
- interpret the value of the constant \(a\),
- interpret the value of the constant \(b\).
- State a long term limitation of the model for \(p\).