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1.7.20 Constructing differential equations (A-level only)

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Question 32

An ecologist is monitoring two distinct populations of insects.

The number of insects, NNN, in the first population is modelled by the equation

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

where A A\,A and k k\,k are positive constants and t t\,t is the time in days from the start of the observation.

Given that:

  • there were 4000 insects in this population at the start of the observation
  • there were 12000 insects 6 days later
a.

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

[4]
b.

The number of insects, NNN, in the second population is modelled by the equation

N=80000e−0.4t,t≥0 N = 80000 e^{-0.4t}, \quad t \ge 0 N=80000e−0.4t,t≥0

where t t\,t is the time in days from the start of the observation.

Find the rate of decrease of insects in this second population exactly 4 days from the start. Give your answer to 3 significant figures.

[3]
c.

When t=Tt = Tt=T, the number of insects in the two populations was the same.

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

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

[4]

1.7.20 Constructing differential equations (A-level only) Questions

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