Respuesta :
We are told that the population decreases 9.6% each year. This means that every year, we keep 100-9.6% of the population. That is, we keep 90.4% of the population of the actual population. The equation that represent this, is
[tex]\text{Pyear = 0.904}\cdot\text{Ppreviousyear}[/tex]Let us number the years referenced to the year in which we have 1000 monkeys. That is, let P0 = 1000. So for year 1 we get
[tex]P_1=0.904\cdot P_0[/tex]For the year two, we have
[tex]P_2=0.904\cdot P_1=0.904\cdot0.904\cdot P_0=(0.904)^2\cdot P_0[/tex]After t years, the population would be
[tex]P_t=(0.904)^t\cdot P_0[/tex]The question is asking what is the value of t if we know that after t years the remaining population is 200 monkeys. So, we have the following equation
[tex]200=(0.904)^t\cdot1000[/tex]If we divide by 1000 on both sides, we get
[tex](0.904)^t=\frac{200}{1000}=\frac{1}{5}[/tex]Now, we apply the natural logarithm on both sides, so we get
[tex]\ln ((0.904)^t)=\ln (\frac{1}{5})[/tex]We will apply this properties of logartithms
[tex]\ln (a^b)=b\ln (a)[/tex][tex]\ln (\frac{a}{b})=\ln (a)-\ln (b)[/tex]So, we get
[tex]t\ln (0.904)=\ln (1)-\ln (5)[/tex]Recall that ln(1)=0. so we get
[tex]t\cdot\ln (0.904)=-\ln (5)[/tex]Finally, we divide by ln(0.904) on both sides. So we get
[tex]t=\frac{-\ln (5)}{\ln (0.904)}[/tex]By using a calculator, we get that t=15.9467. Which means that aproximately in 16 years the population of monkeys will be 200.
