The concentration of a specific chemical, CCC mmol/L, in a bioreactor ttt hours after a growth process starts is modelled by the differential equation
dCdt=α−0.25C \frac{dC}{dt} = \alpha - 0.25C dtdC=α−0.25Cwhere α\alphaα is a positive constant. At the start of the process, there is no trace of the chemical in the reactor.
Solve the differential equation to show that C=4α(1−e−0.25t)C = 4\alpha(1 - e^{-0.25t})C=4α(1−e−0.25t).
In the long term, the concentration in the bioreactor stabilizes at 20 mmol/L.
Find the value of α\alphaα.
Determine the time, in hours, for the concentration to reach 15 mmol/L, giving your answer to 2 significant figures.
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.
