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

This model is not realistic in the long term because it assumes an exponential growth of the damaged plants without accounting for the maximum capacity of the greenhouse, which is only 900 plants. As the model predicts an unlimited increase, it could suggest that all plants would eventually become damaged, which is not feasible since there are only 900 plants in total.

To show this, we rearrange the equation:

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

which implies:

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

Integrating both sides gives:

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

with appropriate selection of $A$ and $B$.

Upon integrating the left side, we find:

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

The solution involves logarithmic forms:

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

Solving for $t$ yields:

$t = \frac{A}{B} \left( \ln(900-x) + \ln(x) \right) + C$

To find the number of days it takes until half the plants (450) are damaged:

We substitute $x = 450$ into the equation for $t$.

After evaluating:

$t = 3(\ln(450) - \ln(900-450)) + 10.67$

This gives the final value of $t$ as approximately 11 days.