Answer:
The population in 40 years will be 1220.
Step-by-step explanation:
The population of a town grows at a rate proportional to the population present at time t.
This means that:
[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.
The initial population of 500 increases by 25% in 10 years.
This means that [tex]P(0) = 500, P(10) = 1.25*500 = 625[/tex]
We apply this to the equation and find t.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]625 = 500e^{10r}[/tex]
[tex]e^{10r} = \frac{625}{500}[/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 40 years
This is P(40).
[tex]P(t) = 500e^{0.0223t}[/tex]
[tex]P(40) = 500e^{0.0223*40} = 1220[/tex]
The population in 40 years will be 1220.
