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Question 680

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

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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