An environmental scientist is monitoring the breakdown of a pollutant in a controlled bioremediation tank. The concentration of the pollutant PPP parts per million (ppm) at time ttt hours after the treatment begins is modelled by the equation:
P=10(5+λe−kt) P = 10(5 + \lambda e^{-kt}) P=10(5+λe−kt)where λ\lambdaλ and kkk are positive constants.
Initially, the concentration of the pollutant is 200200200 ppm. After 7 hours, the concentration falls to 140140140 ppm.
Find the concentration of the pollutant after 24 hours. Give your answer to three significant figures.
Determine the limiting concentration of the pollutant as t→∞t \to \inftyt→∞, giving a reason for your answer.
Find the time taken, in hours, for the concentration to fall to 555 ppm above the limiting concentration.
Explain why the model might need to be adjusted if the remediation process was conducted in an outdoor reservoir.
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.