Write 1(N−20)(N+80)\frac{1}{(N - 20)(N + 80)}(N−20)(N+80)1 in partial fraction form.
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−0.2tHence 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.)
According to the model, the population will eventually fall to 100k100k100k fish.
State the value of the constant kkk.