Question

Due to deforestation in Africa, the population of tana river red Columbus monkeys is decreasing by 9.6 % every year. These are currently only 1000 colobus monkeys left. A) how many years will it be before there are only 200 monkeys left ? B) how many years will it be before there are only 200 monkeys left ?

Answer 1

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

`\text{Pyear = 0.904}\cdot\text{Ppreviousyear}`

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

`P_1=0.904\cdot P_0`

For the year two, we have

`P_2=0.904\cdot P_1=0.904\cdot0.904\cdot P_0=(0.904)^2\cdot P_0`

After t years, the population would be

`P_t=(0.904)^t\cdot P_0`

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

`200=(0.904)^t\cdot1000`

If we divide by 1000 on both sides, we get

`(0.904)^t=(200)/(1000)=(1)/(5)`

Now, we apply the natural logarithm on both sides, so we get

`\ln ((0.904)^t)=\ln ((1)/(5))`

We will apply this properties of logartithms

`\ln (a^b)=b\ln (a)`
`\ln ((a)/(b))=\ln (a)-\ln (b)`

So, we get

`t\ln (0.904)=\ln (1)-\ln (5)`

Recall that ln(1)=0. so we get

`t\cdot\ln (0.904)=-\ln (5)`

Finally, we divide by ln(0.904) on both sides. So we get

`t=(-\ln (5))/(\ln (0.904))`

By using a calculator, we get that t=15.9467. Which means that aproximately in 16 years the population of monkeys will be 200.