CAIE A-Level Mathematics 3.8.4 Interpreting Differential-Equation Models

Practise interpreting differential-equation solutions by substituting contextual values and taking limits to identify predicted times, equilibrium levels and long-run behaviour.

Exam points

substitute a target radius, depth or population into the solved model and calculate its time
take the relevant exponential term to zero as t tends to infinity to find the limiting value
state the contextual meaning of an equilibrium or limiting value rather than only its algebra

[Maximum number: 1]

A large plantation of area $20 \mathrm{~km}^{2}$ is becoming infected with a plant disease. At time t years the area infected is $x \mathrm{~km}^{2}$ and the rate of increase of x is proportional to the ratio of the area infected to the area not yet infected.

When t=0, x=1 and $\frac{\mathrm{d} x}{\mathrm{~d} t}=1$ .

Calculate the value of t at which the entire plantation becomes infected.

All question bank results loaded

3.8.4—Differential equations question 1