A scientist is studying a population of mice on an island - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 4

To derive the rate of growth, we need to differentiate the population model with respect to time t. Starting from:

$N = \frac{900}{3 + 7e^{-0.25t}}$

Using the quotient rule:

$dN = \frac{(0)(3 + 7e^{-0.25t}) - 900(-0.25)(7e^{-0.25t})}{(3 + 7e^{-0.25t})^2} dt$

This simplifies to:

$\frac{dN}{dt} = \frac{900 \cdot 0.25 \cdot 7 e^{-0.25t}}{(3 + 7e^{-0.25t})^2}.$

Now, we know that:

$N = \frac{900}{3 + 7e^{-0.25t}}$

Solving for e^{-0.25t} gives:

$e^{-0.25t} = \frac{900 - 3N}{7N}.$

Substituting this back into the rate of growth expression leads to:

$\frac{dN}{dt} = \frac{N(300 - N)}{1200}$

This confirms the equation for the rate of growth.

To find the time T at which the rate of growth is maximum, we note that it's given that this occurs at 7 months. We can check by setting ( N = 150 ):

From our earlier expression:

$150 = \frac{900}{3 + 7e^{-0.25T}}.$

Rearranging gives:

=> 7e^{-0.25T} = 3 \ => e^{-0.25T} = \frac{3}{7}.$$ Taking natural logarithms: $$-0.25T = \ln(\frac{3}{7}) \ => T = -4 \ln(\frac{3}{7}) = 7 \text{ months}.$$

According to the model, the maximum number of mice on the island is given by:

$P = 300.$

Thus, the answer for P is either 299 or 300.