Answer:
The population will reach 860,000 in 16.67 years from now.
Step-by-step explanation:
The compound growth model is given 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(decimal).
In this problem, we have that:
[tex]P(0) = 430000, r = 0.0425[/tex]
Approximately when will the population reach 860,000?
This is t when [tex]P(t) = 860000[/tex]. So
[tex]P(t) = P(0)(1+r)^{t}[/tex]
[tex]860000 = 430000(1+0.0425)^{t}[/tex]
[tex](1.0425)^{t} = \frac{860000}{430000}[/tex]
[tex](1.0425)^{t} = 2[/tex]
We have the following logarithm rule
[tex]\log{a^{t}} = t\log{a}[/tex]
Applying log to both sides
[tex]\log{(1.0425)^{t}} = \log{2}[/tex]
[tex]t\log(1.0425) = 0.3[/tex]
[tex]0.018t = 0.3[/tex]
[tex]t = \frac{0.3}{0.018}[/tex]
[tex]t = 16.67[/tex]
The population will reach 860,000 in 16.67 years from now.
