Question

A native wolf species has been reintroduced into a national forest. Originally 400 wolves were transplanted, and after 3 years the population had grown to 704 wolves. If the population grows exponentially according to the formula Pt=P0(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? wolves(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. years

Answer 1

A native wolf species has been reintroduced into a national forest.

Originally 400 wolves were transplanted, and after 3 years the population had grown to 704 wolves.

The population grows exponentially according to the formula;

$P_t=P_o(1+r)^t$

So, we can put in the formula, Pt = 704 and Po = 400;

$\begin{gathered} 704=400(1+r)^3 \\ (1+r)^3=1.76 \\ \end{gathered}$

a. We obtain the cube root of both sides as;

$\begin{gathered} 1+r=1.207 \\ r=0.207=20.7\text{ \%} \end{gathered}$

b. In 10 years, t = 10, we would have ;

$P_t=400(1+0.207)^{10}=2625\text{ }$

Therefore, we would have 2625 wolves if the trend continues;

c. In order to find the time to grow to 1000 wolves.

$\begin{gathered} 1000=400(1.207)^t \\ 2.5=1.207^t \\ 0.91629=0.18814t \\ t=4.9 \end{gathered}$

Therefore, it will take about 4.9 years for the population to grow to 1000 wolves.