Respuesta :
Answer:
The alligator population in this region in the year 2020 is estimated to be 18144
Step-by-step explanation:
Malthus population model has the following format:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(0) is the initial population and r is the growth rate.
In 1980 the population of alligators in a particular region was estimated to be 1200.
This means that [tex]P(0) = 1200[/tex]
In 2007 the population had grown to an estimated 7500.
2007 is 27 years after 1980, because 2007-1980 = 27.
So
[tex]P(27) = 7500[/tex]
Replacing this into the equation, allows us to find the value of r.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]7500 = 1200e^{27r}[/tex]
[tex]e^{27r} = \frac{7500}{1200}[/tex]
[tex]e^{27r} = 6.25[/tex]
Applying ln to both sides, which is the inverse operation of the exponential
[tex]\ln{e^{27r}} = \ln{6.25}[/tex]
[tex]27r = 1.8326[/tex]
[tex]r = \frac{1.8326}{27}[/tex]
[tex]r = 0.0679[/tex]
So
[tex]P(t) = 1200e^{0.0679t}[/tex]
The alligator population in this region in the year 2020 is estimated to be
2020 - 1980 = 40
So this is P(40).
[tex]P(t) = 1200e^{0.0679t}[/tex]
[tex]P(40) = 1200e^{0.0679*40} = 18144[/tex]
The alligator population in this region in the year 2020 is estimated to be 18144
