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 Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) 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.
