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1.9.31 Formulate first order differential equations (A-level only)

1.9.31 Formulate first order differential equations (A-level only)

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Question 31

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]
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1.9.31 Formulate first order differential equations (A-level only) Questions

  1. A Level
  2. /Maths
  3. /1.9.31 Formulate first order differential equations (A-level only)

50 exam-style questions on OCR (MEI) A Level Maths 1.9.31 Formulate first order differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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