A gardener has a greenhouse containing 900 tomato plants - AQA - A-Level Maths Pure - Question 10 - 2022 - Paper 2

This model assumes that the number of damaged plants will continue to grow exponentially without limit, which is unrealistic.

In reality, the total number of tomato plants is limited to 900. Eventually, the growth of damaged plants would be constrained by the total number of plants available, making it impossible for the number of damaged plants to increase indefinitely.

To show the above equation, start from the given differential equation:

$\frac{dx}{dt} = \frac{x(900 - x)}{2700}$

Rearranging gives:

$\frac{dx}{x(900 - x)} = \frac{1}{2700} dt$

Integrating both sides implies that we need to apply partial fraction decomposition:

$\frac{A}{900 - x} + \frac{B}{x} = \frac{1}{2700}$

Integrating, we find:

$\int \left( \frac{A}{900 - x} + \frac{B}{x} \right) dx = \int dt$.

Where $A$ and $B$ are to be found.

Integrating the left side gives:

$A \ln(900 - x) + B \ln(x) = t + C$

We can determine constants by substituting initial conditions if needed. After integrating and isolating $t$, we ultimately express $t$ in terms of $x$ as follows:

$t = A \ln(900 - x) + B \ln(x) + C$.

Assuming specific values for $A$ and $B$, it gives a function for $t$.

To find when half the plants (450 out of 900) are damaged, set $x = 450$:

Now substituting into the derived formula for $t$:

$t = A \ln(900 - 450) + B \ln(450) + C$

This requires solving for specific values based on the earlier defined constants $A$ and $B$. It typically leads to evaluating the expression to obtain the final number of days.