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We estimate that the population of a certain, in t years will be given byp (t) = (2t² + 75) / (2t² + 150) million habitantsAccording to this hypothesis:What is the current population?What will it be in the long term?Sketch the population graph

Sagot :

Given that the population can be represented by the equation;

[tex]P(t)=\frac{2t^2+75}{2t^2+150}[/tex]

The current population (Initial population) is the population at time t=0;

Substituting;

[tex]t=0[/tex][tex]\begin{gathered} P(0)=\frac{2t^2+75}{2t^2+150}=\frac{2(0)^2+75}{2(0)^2+150}=\frac{75}{150} \\ P(0)=0.5\text{ million} \end{gathered}[/tex]

Therefore, the current population of the habitat is;

[tex]0.5\text{ million}[/tex]

The long term population would be the population as t tends to infinity;

[tex]\begin{gathered} \lim _{t\to\infty}P(t)=\frac{2t^2+75}{2t^2+150}=\frac{2(\infty)^2+75}{2(\infty)^2+150}=\frac{\infty}{\infty} \\ \lim _{t\to\infty}P(t)=\frac{4t}{4t}=1 \end{gathered}[/tex]

Therefore, the long term population of the habitat is;

[tex]P(\infty)=1\text{ million}[/tex]

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