Suppose a fish population is currently 6000 and 5 years later the population is 18000. Use the explicit exponential model Pn=Po(1+r)^n to find the rate of growth.


The explicit exponential growth model is given by:

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

Where:
- \( P_n \) is the population after \( n \) years,
- \( P_0 \) is the initial population,
- \( r \) is the rate of growth (which we are trying to find),
- \( n \) is the number of years.

Given:
- Initial population \( P_0 = 6000 \),
- Population after 5 years \( P_5 = 18000 \),
- Time \( n = 5 \) years.

### Step 1: Plug in the values into the formula

\[
18000 = 6000 (1 + r)^5
\]

### Step 2: Solve for \( r \)

1. Divide both sides by 6000:

\[
3 = (1 + r)^5
\]

2. Take the fifth root of both sides to get rid of the exponent:

\[
(1 + r) = 3^{\frac{1}{5}}
\]

3. Calculate \( 3^{\frac{1}{5}} \):

\[
1 + r \approx 1.2457
\]

4. Subtract 1 from both sides:

\[
r \approx 0.2457
\]

Thus, the growth rate \( r \) is approximately **0.2457**, or **24.57%**.

Would you like more details on any step or have further questions?

Here are 5 questions to expand on this concept:
1. How can you verify the exponential model results with actual data?
2. What happens to the population growth if the rate \( r \) changes?
3. How would you model a population decline instead of growth?
4. What role does the number of years \( n \) play in exponential models?
5. How can this model apply to real-world scenarios other than population?

**Tip:** In exponential growth, small changes in the growth rate can lead to large differences over time.

Suppose a fish population is currently 6000 and 5 years later the population is 18000. Use the explicit exponential model Pn=Po(1+r)^n to find the rate of growth.

Learn how to calculate the growth rate using the exponential growth model Pn = Po(1 + r)^n. Given a fish population of 6000 and its growth to 18000 in 5 years, this problem demonstrates how to find the rate of growth, step by step. Suitable for students in Grades 9-12.

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