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1.9 Calculus

1.9 Calculus

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Question 392

An environmental scientist is tracking the mass of substances in a filtration tank.

The mass, MMM, in milligrams, of a purifying bacteria culture is modelled by the equation

M=Aekt,t≥0 M = A e^{kt}, \quad t \ge 0 M=Aekt,t≥0

where A A\,A and k k\,k are positive constants and t t\,t is the time in hours since the bacteria were introduced.

Given that:

  • the initial mass of the bacteria was 250 mg
  • after 8 hours, the mass had increased to 1500 mg
a.

Find the exact value of A A\,A and the value of k k\,k correct to 4 significant figures.

[4]
b.

The mass, MMM, of a specific contaminant in the tank is modelled by the equation

M=12000e−0.15t,t≥0 M = 12000 e^{-0.15t}, \quad t \ge 0 M=12000e−0.15t,t≥0

where t t\,t is the time in hours since the start of the filtration process.

Find the rate of decrease of the mass of the contaminant exactly 5 hours after the start. Give your answer in mg per hour to 3 significant figures.

[3]
c.

At time t=Tt = Tt=T, the mass of the bacteria culture is equal to the mass of the contaminant.

Find the value of TTT, giving your answer to 3 significant figures.

(Solutions relying entirely on calculator technology are not acceptable.)

[4]
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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