Respuesta :
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]
The size of the population after t years will be P(t) = 160000/1920e^-1.1856t + 80.
How to calculate the population
Based on the information given, the following can be noted:
Po = Initial population = 80
N = carrying capacity = 2000
P(t) = (80 × 2000)/(2000 - 80)e^-rt + 80
= 160000/(1920)e^-rt + 80
Since the fish population tripled in the first year, the population will be:
P(1) = 80 × 3 = 240
240 = 160000/1920e^-r + 80
1920e^-r + 80 = 2000/3
e^-r = [(2000/3) - 80]/1920
e^-r = 0.3056
-r = In(0.3056)
r = 1.1856
Therefore, the population after t years will be:
P(t) = 160000/1920e^-1.1856t + 80
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