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1.9 Calculus

1.9 Calculus

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Question 334

The pressure, PPP kPa, inside a leaking decompression chamber ttt minutes after the leak is first detected, is modelled by the differential equation

dPdt=−k(P−100)2 \frac{\text{d}P}{\text{d}t} = -k(P - 100)^2 dtdP​=−k(P−100)2

where kkk is a positive constant and P>100P > 100P>100.

Given that the pressure in the chamber:

  • is 110011001100 kPa at the instant the leak is detected
  • is 350350350 kPa exactly 101010 minutes later
a.

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

P=at+bct+d P = \frac{at + b}{ct + d} P=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 pressure in the chamber to reach 150150150 kPa. Give your answer to the nearest minute.

[3]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret 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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