Respuesta :
In order to calculate the amount of time needed, we can use the formula below:
[tex]P=P_0\cdot(1+r)^{\frac{t}{n}}[/tex]Where P is the population after t years, P0 is the initial population, r is the rate and n is the period of half-life.
So, for P = 0.1*P0, r = -0.5 (the population decreases by half its amount) and n = 25, we have:
[tex]\begin{gathered} 0.1P_0=P_0(1-0.5)^{t/25} \\ 0.1=0.5^{t/25} \\ \log (0.1)=\log (0.5^{t/25}) \\ \log (0.1)=\frac{t}{25}\log (0.5) \\ \frac{t}{25}=\frac{\log (0.1)}{\log (0.5)} \\ \frac{t}{25}=\frac{-1}{-0.301} \\ \frac{t}{25}=3.322 \\ t=83 \end{gathered}[/tex]Therefore it will take approximately 83 years.
