13.
Figure 5
A colony of ants is being studied. The number of ants in the colony is modelled by the equation
$$P = 200 - \frac { 160 \mathrm { e } ^ { 0.6 t } } { 15 + \mathrm { e } ^ { 0.8 t } } \quad t \in \mathbb { R } , t \geqslant 0$$
where \(P\) is the number of ants, measured in thousands, \(t\) years after the study started. A sketch of the graph of \(P\) against \(t\) is shown in Figure 5
- Calculate the number of ants in the colony at the start of the study.
- Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
The population of ants initially decreases, reaching a minimum value after \(T\) years, as shown in Figure 5
- Using your answer to part (b), calculate the value of \(T\) to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)