2025 Nov TZ1 P2 Q8

A population of frogs, \(F\), in a swamp after \(t\) months, can be modelled by the function

\(F(t) = 1850 \times 1.105^t\) where \(t \geq 0\).

1. Find the population of frogs after one year.

   [2]

2. After \(x\) complete months, the population will be at least \(35 \ 000\) frogs. Find the value of \(x\).

   [3]

The function \(F\) can be written in the form \(F(t) = 1850e^{kt}\).

3. Find the exact value of \(k\).

   [2]

4. Find the rate at which the population of frogs is growing after \(15\) months.

   [2]

A more realistic model describing the population of frogs, \(G\), after \(t\) months is given by

\(G(t) = \frac{3500}{1 + Ae^{-0.0998t}}\) where \(t \geq 0\).

5. After \(15\) months, this model predicts a population of \(6995\) frogs. Find the value of \(A\).

   [2]

6. Find the value of \(t\) when the rate of population growth is the greatest.

   [2]

7. By considering the graph of \(G\) or otherwise, state one reason why \(G(t)\) is a more appropriate long-term model than \(F(t)\).

   [1]