A biological culture is treated with a disinfectant. The population of bacteria, PPP, in thousands, decreases at a rate modeled in terms of its current population by the equation:
Rate of decrease=0.4P2 \text{Rate of decrease} = 0.4P^2 Rate of decrease=0.4P2Initially, the population of the culture is 5000 bacteria, so P=5P = 5P=5 at t=0t = 0t=0, where ttt is the time in hours after the disinfectant is applied.
By first forming a suitable differential equation involving dPdt\frac{dP}{dt}dtdP, show that
P=52t+1 P = \frac{5}{2t + 1} P=2t+15Determine the rate of change of the population when t=2t = 2t=2.
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.