Respuesta :
The bacteria population is given by the following formula:
[tex]P(t)=P_0(r)^t[/tex]where P0 represents the initial population and r the increasing/decreasing rate and t represents the time. The initial population is given:
[tex]P_0=80000[/tex]The population grows by 10% every 2 years, therefore, the current population is multiplied by 1.1 every two years. If we consider the unit of time as one year, then, the equation for our bacteria population is:
[tex]P(t)=80000(1.1)^{t/2}[/tex]We want to find the corresponding value for t when P is equal to 25000.
[tex]25000=80000(1.1)^{t/2}[/tex]Solving for t, we have:
[tex]\begin{gathered} 25000=80000(1.1)^{t/2} \\ \frac{25000}{80000}=1.1^{t/2} \\ \frac{5}{16}=1.1^{t/2} \\ \log\frac{5}{16}=\log1.1^{t/2} \\ \log5-\log16=\frac{t}{2}\log1.1 \\ t=2(\frac{\log5-\log16}{\log1.1}) \\ t=2\log_{1.1}\frac{5}{16} \\ t=-12.2038466... \end{gathered}[/tex]