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1.7 Differentiation

1.7 Differentiation

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Question 202

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

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.

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