The current, xxx amps, at time ttt seconds after a switch is closed in an electric circuit is modelled by the differential equation
dxdt=k−2x \frac{dx}{dt} = k - 2x dtdx=k−2xwhere kkk is a constant. Initially, there is no current in the circuit.
Solve the differential equation to show that x=k2(1−e−2t)x = \frac{k}{2}(1 - e^{-2t})x=2k(1−e−2t).
In the long term, the current in the circuit approaches 12 amps.
Find the value of kkk.
Find the time, in seconds, for the current to reach 8 amps, 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.
