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1.11 H: Integration

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Question 45

The concentration, CCC mg/L, of a particular chemical residue in a large processing tank, ttt hours after a neutralising catalyst is introduced, is modelled by the differential equation

dCdt=−k(C−10)2 \frac{\text{d}C}{\text{d}t} = -k(C - 10)^2 dtdC​=−k(C−10)2

where kkk is a constant and C>10C > 10C>10.

Given that the concentration of the residue:

  • is 610610610 mg/L at the instant the catalyst is added
  • is 110110110 mg/L exactly 222 hours after the catalyst is added
a.

Solve the differential equation to show that, according to the model

C=at+bct+d C = \frac{at + b}{ct + d} C=ct+dat+b​

where a,b,ca, b, ca,b,c and ddd are integers to be found.

[6]
b.

Hence find, according to the model, the time taken for the concentration of the residue to fall to 404040 mg/L. Give your answer to the nearest hour.

[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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