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1.7.20 Constructing differential equations (A-level only)

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Question 3
a.

Write 1(N−20)(N+80)\frac{1}{(N - 20)(N + 80)}(N−20)(N+80)1​ in partial fraction form.

[3]
b.

An invasive species of fish is being removed from a large lake to protect the local ecosystem. The population of this species, NNN (measured in hundreds), is modelled by the differential equation

dNdt=−(N−20)(N+80)500 \frac{dN}{dt} = -\frac{(N - 20)(N + 80)}{500} dtdN​=−500(N−20)(N+80)​

where ttt is the time, in years, from when the removal program began.

Given that the initial population of the species was 12,00012,00012,000 fish (so N=120N = 120N=120 at t=0t = 0t=0),

solve the differential equation to show that

N=40+80e−0.2t2−e−0.2t N = \frac{40 + 80e^{-0.2t}}{2 - e^{-0.2t}} N=2−e−0.2t40+80e−0.2t​
[6]
c.

Hence find the time taken for the population of the species to fall to 4,5004,5004,500 fish.

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

[3]
d.

According to the model, the population will eventually fall to 100k100k100k fish.

State the value of the constant kkk.

[1]

1.7.20 Constructing differential equations (A-level only) Questions

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