8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation
$$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$
- Calculate the number of primroses at the start of the study.
- Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
- Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
- Explain why the population of primroses can never be 270