Problem Statement
The population $P(t)$ of trout introduced into a lake is modeled by the logistic equation:
$$P(t) = \frac{4800}{1 + 7 e^{-0.25 t}}, \quad t \ge 0$$
where $t$ is measured in months.
**(a)** [2 marks] Find the initial population of trout introduced into the lake.
**(b)** [2 marks] Write down the carrying capacity of the lake.
**(c)** [3 marks] Find the time $t$ when the population is increasing at its maximum rate.
Verified Solution & Marking Scheme
Part (a) [M1]
Substitute $t = 0$: $P(0) = \frac{4800}{1 + 7e^0} = \frac{4800}{8}$
Part (a) [A1]
Initial population is $600$ trout.
Part (b) [M1, A1]
As $t \to \infty$, $e^{-0.25t} \to 0$. Carrying capacity is $4800$ trout.
Part (c) [M1]
Maximum growth rate occurs at inflection point, where population is half the carrying capacity: $P(t) = \frac{4800}{2} = 2400$
Part (c) [M1]
$\frac{4800}{1 + 7e^{-0.25t}} = 2400 \implies 1 + 7e^{-0.25t} = 2 \implies 7e^{-0.25t} = 1 \implies e^{-0.25t} = \frac{1}{7}$
Part (c) [A1]
$-0.25t = \ln(1/7) = -\ln 7 \implies t = 4\ln 7 \approx 7.78\text{ months}$