A chemical reaction produces a substance that spreads across a filter paper. The area of the paper, A cm2A\text{ cm}^2A cm2, covered by the substance ttt hours after the reaction starts is modeled by the differential equation
dAdt=A324t2,t>0 \frac{\text{d}A}{\text{dt}} = \frac{A^{\frac{3}{2}}}{4t^2}, \quad t > 0 dtdA=4t2A23,t>0Given that A=4A = 4A=4 when t=2t = 2t=2,
show that
A=(ptqt+r)2 A = \left( \frac{pt}{qt + r} \right)^2 A=(qt+rpt)2where ppp, qqq, and rrr are integers to be found.
According to the model, find the limiting value of the area covered as t→∞t \to \inftyt→∞.
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.
