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1.11 H: Integration

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Question 86

An industrial spill of a viscous lubricant is spreading across a flat floor. The area of the spill, L m2L \text{ m}^2L m2, at time ttt hours after the spill is discovered, is modelled by the differential equation

dLdt=LL5t2,t>0 \frac{dL}{dt} = \frac{L\sqrt{L}}{5t^2}, \quad t > 0 dtdL​=5t2LL​​,t>0

Given that the spill covers 16 m216 \text{ m}^216 m2 at the moment it is discovered (t=1t = 1t=1),

a.

show that

L=(ptqt+r)2 L = \left( \frac{pt}{qt + r} \right)^2 L=(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 area covered by the spill 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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