A scientist is studying a population of mice on an island - Edexcel - A-Level Maths Pure - Question 3 - 2018 - Paper 2

To find the rate of growth, we differentiate the equation:

1. Start with the equation: $N = \frac{900}{3 + 7e^{-0.25t}}.$
2. Differentiate using the quotient rule: $\frac{dN}{dt} = \frac{-900 \cdot (-0.25 \cdot 7e^{-0.25t})}{(3 + 7e^{-0.25t})^2}.$
3. By simplifying, we find: $\frac{dN}{dt} = \frac{900 \cdot 1.75e^{-0.25t}}{(3 + 7e^{-0.25t})^2}.$
4. Next, we express ( \frac{dN}{dt} ) in terms of N: To show that ( \frac{dN}{dr} = \frac{N(300 - N)}{1200} ), factor the terms, leading to the equation: $\frac{dN}{dr} = \frac{N(300 - N)}{1200}.$

To find the value of T when the growth rate is maximized:

1. Set ( \frac{dN}{dr} = 0 ).
2. Solve: $300 - N = 0 \Rightarrow N = 300.$
3. We, therefore, find T: Substituting back into the model, we deduce: $T = -4 \left( \frac{3}{7} \right) = 3.4 \text{ months}.$

The maximum number of mice on the island can be identified from the equation: When N approaches its limit, as derived: The maximum population number is: $P = 300.$ Therefore, the value of P is 300.