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1.11 H: Integration

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Question 203

A crystalline deposit of mass MMM milligrams is formed on an electrode during a chemical process. At time ttt seconds after the process begins, the rate of increase of the mass is modeled by the differential equation

dMdt=M326t2,t>0 \frac{\text{d}M}{\text{dt}} = \frac{M^{\frac{3}{2}}}{6t^2}, \quad t > 0 dtdM​=6t2M23​​,t>0

Given that M=16M = 16M=16 when t=4t = 4t=4,

a.

show that

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

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

[6]
b.

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

[2]
Markscheme

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

480 exam-style questions on AQA A Level Maths 1.11 H: Integration, 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). Each one has a worked solution and a mark scheme showing where the marks go.

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