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Question 644

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>0

Given that A=4A = 4A=4 when t=2t = 2t=2,

a.

show that

A=(ptqt+r)2 A = \left( \frac{pt}{qt + r} \right)^2 A=(qt+rpt​)2

where ppp, qqq, and rrr are integers to be found.

[5]
b.

According to the model, find the limiting value of the area covered as t→∞t \to \inftyt→∞.

[2]
Markscheme

Integration Questions

  1. A Level
  2. /Maths
  3. /Integration

864 exam-style questions on Edexcel A Level Maths Integration, 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. Each one has a worked solution and a mark scheme showing where the marks go.

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