2026 GESS P2 Q4

A rainforest started with a population of 27 deers. The population of the deers is predicted to increase so that after a period of \(t\) years, the population, \(P_f\), is given by the equation \(P_f=\dfrac{a}{3+2e^{-0.18t}}\), where \(a\) is a constant to be found.

1. Show that \(a=135\).[1]
2. Find the minimum number of years required for the population of the deers to increase to 35 deers.[3]
3. Another population of deers lives in a national park. The population, \(P_n\), is given by the equation \(P_n=\dfrac{135}{3+2e^{\lambda t}}\).
   1. The population of deers increase to 40 after 5 years. Find \(\lambda\).[3]
   2. Find the value of \(P_n\) as \(t\) becomes very large.[1]