Respuesta :
Answer:
The population in 20 years will be 781.
Step-by-step explanation:
The population of the town can be modeled by the following equation:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate, as a decimal.
The initial population of 500 increases by 25% in 10 years.
This means that [tex]P(0) = 500, P(10) = 1.25*P(0) = 625[/tex]
We use this to find r.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]625 = 500e^{10r}[/tex]
[tex]e^{10r} = 1.25[/tex]
Applying ln to both sides
[tex]\ln{e^{10r}} = \ln{1.25}[/tex]
[tex]10r = \ln{1.25}[/tex]
[tex]r = \frac{\ln{1.25}}{10}[/tex]
[tex]r = 0.0223[/tex]
So
[tex]P(t) = 500e^{0.0223t}[/tex]
What will be the population in 20 years
This is P(20)
[tex]P(20) = 500e^{0.0223*20} = 781[/tex]
The population in 20 years will be 781.
