Respuesta :
As suggested, first, we will construct a model and then we will evaluate it at t=10, t=15, and finally, we will set the equation equal to 100,000 and solve for t.
The species grows 39% every year, therefore, if T is the total number of beetles, we can set the following equation:
[tex]T=100(1+0.39)^t.[/tex]Evaluating the above equation at t=10, we get:
[tex]T=100(1.39)^{10}=2692.452204\approx2692.[/tex]Evaluating the above equation at t=15, we get:
[tex]T=100(1.39)^{15}=13970.82343\approx13971.[/tex]Note that I rounded to the nearest integer because you cannot have a piece of beetle.
Now, setting T=100,000 and solving for t, we get:
[tex]\begin{gathered} 100,000=100(1.39)^t, \\ 1000=1.39^t, \\ \log 1000=t\log 1.39, \\ t=\frac{\log 1000}{\log 1.39}=\frac{3}{\log 39}\approx21. \end{gathered}[/tex]Answer:
After 10 years there will be 2692 beetles.
After 15 years there will be 13971 beetles.
About 21 years later there will be 100,000 beetles.
