Respuesta :
Answer:
It will take the 20 years for the population to quadruple.
Step-by-step explanation:
Exponential population growth:
An exponential model for population growth has the following model:
[tex]P(t) = P(0)(1+r)^t[/tex]
In which P(0) is the initial population and r is the growth rate, as a decimal.
It grows with a doubling time of 10 years.
This means that [tex]P(10) = 2P(0)[/tex]. We use this to find r. So
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]2P(0) = P(0)(1+r)^10[/tex]
[tex](1+r)^10 = 2[/tex]
[tex]\sqrt[10]{(1+r)^10} = \sqrt[10]{2}[/tex]
[tex]1+r = 2^{\frac{1}{10}}[/tex]
[tex]1 + r = 1.0718[/tex]
So
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]P(t) = P(0)(1.0718)^t[/tex]
Determine how long it will take for the population to quadruple.
This is t for which P(t) = 4P(0). So
[tex]P(t) = P(0)(1.0718)^t[/tex]
[tex]4P(0) = P(0)(1.0718)^t[/tex]
[tex](1.0718)^t = 4[/tex]
[tex]\log{(1.0718)^t} = \log{4}[/tex]
[tex]t\log{1.0718} = \log{4}[/tex]
[tex]t = \frac{\log{4}}{\log{1.0718}}[/tex]
[tex]t = 20[/tex]
It will take the 20 years for the population to quadruple.
