9. A rare species of mammal is being studied. The population \(P\), \(t\) years after the study started, is modelled by the formula
$$P = \frac { 900 \mathrm { e } ^ { \frac { 1 } { 4 } t } } { 3 \mathrm { e } ^ { \frac { 1 } { 4 } t } - 1 } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$
Using the model,
- calculate the number of mammals at the start of the study,
- calculate the exact value of \(t\) when \(P = 315\)
Give your answer in the form \(a \ln k\), where \(a\) and \(k\) are integers to be determined.
- Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\)
- Hence find the value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 8\), giving your answer to 2 decimal places.