Answer:
46 years
Step-by-step explanation:
We have the logistic growth function [tex]f(t)=\frac{25,000}{1+8.25e^{-0.076t}}[/tex] and we want to find the time when the population will reach 20,000, to do it we just need to replace [tex]f(x)[/tex] with 20,000 and solve for [tex]t[/tex]:
[tex]f(t)=\frac{25,000}{1+8.25e^{-0.076t}}[/tex]
[tex]20,000=\frac{25,000}{1+8.25e^{-0.076t}}[/tex]
Divide both sides by 25,000
[tex]\frac{20,000}{25,000} =\frac{1}{1+8.25e^{-0.076t}}[/tex]
[tex]0.8=\frac{1}{1+8.25e^{-0.076t}}[/tex]
Multiply both sides by [tex]1+8.25e^{-0.076t}[/tex] and divide them by 0.8
[tex]1+8.25e^{-0.076t}=1.25[/tex]
Subtract 1 from both sides
[tex]8.25e^{-0.076t}=0.25[/tex]
Divide both sides by 8.25
[tex]e^{-0.076t}=\frac{0.25}{8.25}[/tex]
[tex]e^{-0.076t}=\frac{1}{33}[/tex]
Take natural logarithm to both sides
[tex]ln(e^{-0.076t})=ln(\frac{1}{33} )[/tex]
[tex]-0.076t=ln(\frac{1}{33} )[/tex]
Divide both sides by -0.076
[tex]t=\frac{ln(\frac{1}{33} )}{-0.076}[/tex]
[tex]t[/tex] ≈ 46
We can conclude that the population will reach 20,000 after 46 years.
