Exponential and logistic models for a rabbit population, growth rate and the time of fastest growth

Rabbits are introduced to an island. The number of rabbits, \(N\), \(t\) months after they arrive is modelled by

\(N(t)=240\times1.08^{t}\), \(t\geq0\).

1. Find the number of rabbits two years after they arrive.[2]
2. The number of rabbits first reaches at least \(5000\) after \(m\) complete months. Find \(m\).[3]

The model can also be written as \(N(t)=240e^{kt}\).

3. Find the exact value of \(k\).[2]
4. Find the rate at which the number of rabbits is increasing \(10\) months after they arrive.[2]

An ecologist suggests a second model for the number of rabbits, \(M\), \(t\) months after they arrive:

\(M(t)=\dfrac{6000}{1+Ae^{-0.077t}}\), \(t\geq0\).

5. This model gives \(480\) rabbits \(10\) months after they arrive. Find \(A\).[2]
6. Find the value of \(t\) at which, according to this model, the number of rabbits is increasing most rapidly.[2]
7. With reference to the long-term behaviour of the graph of \(M\), give one reason why the second model is more suitable than the first for predicting the number of rabbits many years after they arrive.[1]