Respuesta :
Answer:
24.5 years will be required for the population to double its initial value.
Step-by-step explanation:
The population of Waterville can be modeled by the following equation.
[tex]P(t) = P_{0}(1 + r)^{t}[/tex]
In which [tex]P_{0}[/tex] is the initial population and r is the growth rate.
The population of Waterville increased 12% during 4 years.
This means that [tex]P(4) = 1.12P_{0}[/tex]
With this, we can find r
[tex]P(t) = P_{0}(1 + r)^{t}[/tex]
[tex]1.12P_{0} = P_{0}(1 + r)^{4}[/tex]
[tex](1+r)^{4} = 1.12[/tex]
Applying the fourth root to both sides
[tex]1 + r = 1.0287[/tex]
So
[tex]P(t) = P_{0}(1.0287)^{t}[/tex]
How many years are required for the population to double its initial value?
This is t when [tex]P(t) = 2P_{0}[/tex]
So
[tex]P(t) = P_{0}(1.0287)^{t}[/tex]
[tex]2P_{0} = P_{0}(1.0287)^{t}[/tex]
[tex](1.0287)^{t} = 2[/tex]
How we find t?
Logarithims
We have that
[tex]\log{a}^{t} = t*\log{a}[/tex]
So we apply log to both sides
[tex]\log{(1.0287)^{t}} = \log{2}[/tex]
[tex]t\log{1.0287} = \log{2}[/tex]
[tex]t = \frac{\log{2}}{\log{1.0287}}[/tex]
[tex]t = 24.5[/tex]
24.5 years will be required for the population to double its initial value.
