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1.11 H: Integration

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Question 107

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.

a.

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<c

where ccc is a constant to be found that represents the time when the population first returns to its initial value.

[5]
b.

Find the exact maximum population predicted by this model. Fully justify that your answer is a maximum.

[5]
Markscheme

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

480 exam-style questions on AQA A Level Maths 1.11 H: Integration, 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). Each one has a worked solution and a mark scheme showing where the marks go.

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