Respuesta :
Answer:
The initial population was approximatedly 3535 inhabitants.
Step-by-step explanation:
The population of the city can be given by the following differential equation.
[tex]\frac{dP}{dt} = Pr[/tex],
In which r is the rate of growth of the population.
We can solve this diffential equation by the variable separation method.
[tex]\frac{dP}{dt} = Pr[/tex]
[tex]\frac{dP}{P} = r dt[/tex]
Integrating both sides:
[tex]ln P = rt + c[/tex]
Since ln and the exponential are inverse operations, to write P in function of t, we apply ln to both sides.
[tex]e^{ln P} = e^{rt + C}[/tex]
[tex]P(t) = Ce^{rt}[/tex]
C is the initial population, so:
[tex]P(t) = P(0)e^{rt}[/tex]
Now, we apply the problem's statements to first find the growth rate and then the initial population.
The problem states that:
In two years the population has doubled:
[tex]P(2) = 2P(0)[/tex]
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]2P(0) = P(0)e^{2r}[/tex]
[tex]2 = e^{2r}[/tex]
To isolate r, we apply ln both sides
[tex]e^{2r} = 2[/tex]
[tex]ln e^{2r} = ln 2[/tex]
[tex]2r = 0.69[/tex]
[tex]r = \frac{0.69}{2}[/tex]
[tex]r = 0.3466[/tex]
So
[tex]P(t) = P(0)e^{0.3466t}[/tex]
In two years the population has doubled and a year later there were 10,000 inhabitants.
[tex]P(3) = 10,000[/tex]
[tex]P(t) = P(0)e^{0.3466t}[/tex]
[tex]10,000= P(0)e^{0.3466*3}[/tex]
[tex]P(0) = \frac{10,000}{e^{1.04}}[/tex]
[tex]P(0) = 3534.55[/tex]
The initial population was approximatedly 3535 inhabitants.
