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11.11 Modelling with Differential Equations

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Question 12

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

11.11 Modelling with Differential Equations Questions

  1. A Level
  2. /Maths
  3. /11.11 Modelling with Differential Equations

50 exam-style questions on Edexcel A Level Maths 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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