Answer:
[tex]P(t) = P(0)e^{0.0693t}[/tex]
Step-by-step explanation:
The population of a community is known to increase at a rate proportional to the number of people present at time t.
This means that the population growth is modeled by the following differential equation:
[tex]\frac{dP(t)}{dt} = rP(t)[/tex]
Which has the following solution:
[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 population has doubled in 10 years
This means that [tex]P(10) = 2P(0)[/tex]. We use this to find r.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]2P(0) = P(0)e^{10r}[/tex]
[tex]e^{10r} = 2[/tex]
[tex]\ln{e^{10r}} = \ln{2}[/tex]
[tex]10r = \ln{2}[/tex]
[tex]r = \frac{\ln{2}}{10}[/tex]
[tex]r = 0.0693[/tex]
So the equation that will estimate the population of the community in t years is:
[tex]P(t) = P(0)e^{0.0693t}[/tex]
