Biologists stocked a lake with 80 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 2,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. P =

Answer:[tex]P(t) = \frac{160000e^{1.36t}}{2000 + 80(e^{1.36t} - 1)}[/tex]Step-by-step explanation:The logistic equation is the following one:[tex]P(t) = \frac{KP(0)e^{rt}}{K + P(0)(e^{rt} - 1)}[/tex]In which P(t) is the size of the population after t years, K is the carrying capacity of the population, r is the decimal growth rate of the population and P(0) is the initial population of the lake.In this problem, we have that:Biologists stocked a lake with 80 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 2,000. This means that [tex]P(0) = 80, K = 2000[/tex].The number of fish tripled in the first year. This means that [tex]P(1) = 3P(0) = 3(80) = 240[/tex].Using the equation for P(1), that is, P(t) when [tex]t = 1[/tex], we find the value of r.[tex]P(t) = \frac{KP(0)e^{rt}}{K + P(0)(e^{rt} - 1)}[/tex][tex]240 = \frac{2000*80e^{r}}{2000 + 80(e^{r} - 1)}[/tex][tex]280*(2000 + 80(e^{r} - 1)) = 160000e^{r}[/tex][tex]280*(2000 + 80e^{r} - 80) = 160000e^{r}[/tex][tex]280*(1920 + 80e^{r}) = 160000e^{r}[/tex][tex]537600 + 22400e^{r} = 160000e^{r}[/tex][tex]137600e^{r} = 537600[/tex][tex]e^{r} = \frac{537600}{137600}[/tex][tex]e^{r} = 3.91[/tex]Applying ln to both sides.[tex]\ln{e^{r}} = \ln{3.91}[/tex][tex]r = 1.36[/tex]This means that the expression for the size of the population after t years is:[tex]P(t) = \frac{160000e^{1.36t}}{2000 + 80(e^{1.36t} - 1)}[/tex]