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) The growth rate is 20.7%.

(b) If this trend continues, there will be 2625.0 wolves in ten years.

(c) If this trend continues, it will take 4.9 years for the population to grow to 1000 wolves.

In Mathematics and Statistics, a mathematical model for any population or material that increases by "r" percent per unit of time is an exponential function of this form:

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

Where:

• P(t) represents the future value.
• P represents the initial value.
• b represents the rate of change or growth rate.
• x represents the time.

Based on the information provided above, the growth rate can be calculated as follows;

`704 = 400(1 + r)^(3)\\\\(704)/(400) =(1 + r)^(3)\\\\1.76=(1 + r)^(3)\\\\\sqrt[3]{1.76} =1+r\\\\`

r = 1.207 - 1

r = 0.207 × 100

r = 20.7%.

Part b.

In ten years, the number of wolves is given by;

`P(t) = P(1 + r)^(t)\\\\P(10) = 400(1 + 0.207)^(10)\\\\P(10) = 400(1.207)^(10)`

P(10) = 2625.0 wolves.

Part c.

For the number of years it would take, we have;

`P(t) = P(1 + r)^(t)\\\\1000 = 400(1 + 0.207)^(t)\\\\2.5 = (1.207)^(t)\\\\ln(2.5)=ln(1.207)t\\\\t=(ln(2.5))/(ln(1.207))`

t = 4.87 ≈ 4.9 years.

Answer 2

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.