Question

A native wolf species has been reintroduced into a national forest. Originally 200 wolves were transplanted, and after 3 years the population had grown to 781 wolves. If the population grows exponentially according to the formula P(t)=P(O)(1+r)^t(a) Find the growth rate. Round your answer to the nearest tenth of a percent.r = %(b) If this trend continues, how many wolves will there be in ten years?(c) If this trend continues, how long will it take for the population to grow to 1000 wolves? Round your answer to the nearest tenth of a year.

Answer 1

Step-by-step explanation:

we know population growth is an exponential phenomenon. Therefore, the sequence formed will be a geometric sequence. If P_0 is the first term and P_1 is the successive term, then we get:

`P_1=P_0(1+r)`

where r is the common ratio or the growth rate. Applying the data of the problem to the previous equation, we get:

`781=200(1+r)`

this is equivalent to:

`781\text{ = 200 +200r}`

solving for 200r, we get:

`200r\text{ = 781 -200=581}`

solving for r, we obtain:

`r=(581)/(200)=2.9`

Now, if the population grows exponentially, this trend can be modeled using the following formula:

`P(t)=P_0(1+r)^t`

Applying the obtained r value and the data of the problem to the previous equation, we get:

`P(t)=200(1+2.9)^t`

this is equivalent to:

`P(t)=200(3.9)^t`

with this formula, we can calculate the number of wolves in 10 years:

`P(10)=200(3.9)^(10)=162808121.7`

Now, if the population grows to 1000 wolves, we get the following equation:

`1000=200(3.9)^t`

Solving for t, we get:

`t=\text{ 1.18}\approx1.2\text{ years}`

We can conclude that the correct answer is:

Answer:

a) r = 2.9 ( as a percent would be 290%)

b) 162808121.7

c) 1.2 years