The concentration of a cleaning agent, CCC grams per litre, in a mixing tank at time ttt minutes after the supply is activated is modelled by the differential equation
dCdt=R−0.15C \frac{dC}{dt} = R - 0.15C dtdC=R−0.15Cwhere RRR is a constant. Initially, the tank contains pure water.
Solve the differential equation to show that C=R0.15(1−e−0.15t)C = \frac{R}{0.15}(1 - e^{-0.15t})C=0.15R(1−e−0.15t).
In the long term, the concentration of the cleaning agent in the tank approaches 40 grams per litre.
Find the value of RRR.
Find the time, in minutes, for the concentration to reach 25 grams per litre, 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.
