A native wolf species has been reintroduced into a national forest. Originally 250 wolves were transplanted, and after 3 years the population had grown to 695 wolves. If the population
grows exponentially according to the formula Pn = Po(1 + r)~
(a) Find the growth rate. Do not write the percent % symbol.
Write growth rate as a percent to 3 decimal places.
「=
(b) If this trend continues, how many wolves will there be in ten years?
Round down to the nearest wolf.
wolves
(c) If this trend continues, how long will it take for the population to grow to 1000 wolves?
Round to the nearest tenth of a year.

We are given the exponential growth formula for population growth:

\[
P_n = P_0(1 + r)^n
\]

Where:
- \( P_n \) is the population after \( n \) years,
- \( P_0 \) is the initial population,
- \( r \) is the growth rate, and
- \( n \) is the number of years.

### Part (a) - Finding the Growth Rate

We know that:
- \( P_0 = 250 \) (initial population),
- \( P_3 = 695 \) (population after 3 years).

We can plug these into the exponential growth formula:

\[
695 = 250(1 + r)^3
\]

Solving for \( r \):
\[
\frac{695}{250} = (1 + r)^3
\]
\[
2.78 = (1 + r)^3
\]

Now take the cube root of both sides:

\[
(1 + r) = \sqrt[3]{2.78}
\]

Let’s calculate \( r \).

The growth rate \( r \) is \( 40.610\% \).

### Part (b) - Population in 10 Years

We now use the same formula to calculate the population after 10 years. We know the growth rate \( r \) and the initial population \( P_0 = 250 \). We need to find \( P_{10} \).

\[
P_{10} = 250(1 + 0.4061)^{10}
\]

Let me calculate this.The projected population of wolves after 10 years is approximately **7,552** wolves.

### Part (c) - Time to Reach 1,000 Wolves

We need to determine the time \( n \) for the population to reach 1,000 wolves. We use the formula:

\[
1000 = 250(1 + 0.4061)^n
\]

Now, solve for \( n \):

\[
\frac{1000}{250} = (1 + 0.4061)^n
\]
\[
4 = (1.4061)^n
\]

Taking the natural logarithm on both sides:

\[
\ln(4) = n \ln(1.4061)
\]

Let's calculate \( n \).It will take approximately **4.1 years** for the wolf population to grow to 1,000 wolves.

---

Would you like further details on any of the parts? Here are some additional questions related to the topic:

1. How would the growth rate change if the population reached 1,000 wolves in 3 years instead of 4.1?
2. What would happen to the population if the growth rate decreased by 10%?
3. How would the population size differ if the initial number of wolves was 500 instead of 250?
4. Can you calculate the population after 20 years with the same growth rate?
5. How does exponential growth differ from linear population growth?

**Tip:** Exponential growth often starts slow but accelerates rapidly as time passes due to the compounding effect of growth on larger populations.

A native wolf species has been reintroduced into a national forest. Originally 250 wolves were transplanted, and after 3 years the population had grown to 695 wolves. If the population grows exponentially according to the formula Pn = Po(1 + r)^n:
(a) Find the growth rate. Do not write the percent % symbol. Write growth rate as a percent to 3 decimal places.
(b) If this trend continues, how many wolves will there be in ten years? Round down to the nearest wolf.
(c) If this trend continues, how long will it take for the population to grow to 1000 wolves? Round to the nearest tenth of a year.

Learn how to calculate the growth rate and project future populations in an exponential growth model using a wolf population example. Given an initial population of 250 wolves, discover how to calculate growth rates, future populations, and the time it takes to reach 1,000 wolves. Ideal for high school students studying exponential growth and logarithms.

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