A biologist is studying the population, PPP, of a specific strain of bacteria in a petri dish. The rate of change of the population is modeled by the differential equation
dPdt=3P(4−t)8 \frac{dP}{dt} = \frac{3P(4 - t)}{8} dtdP=83P(4−t)where t≥0t \ge 0t≥0 is the time in hours since the start of the experiment. Initially, the population is 40 units.
Solve the differential equation to show that the population at time ttt is given by
P=40e316(8t−t2)for 0<t<c P = 40 e^{\frac{3}{16}(8t - t^2)} \quad \text{for } 0 < t < c P=40e163(8t−t2)for 0<t<cwhere ccc is a constant to be found that represents the time when the population first returns to its initial value.
Find the exact maximum population predicted by this model. Fully justify that your answer is a maximum.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
