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 Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) 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.
