Suppose a fish population is currently 4000 and 5 years later the population is 10000. Use the explicit exponential model Pn=Po(1+r)^n, to find the rate of growth.

Express the rate of growth as a percentage to the nearest tenth.

The exponential growth model is given by the formula:

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

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

### Step 1: Identify the known values
- Initial population \(P_0 = 4000\),
- Population after 5 years \(P_n = 10000\),
- Time \(n = 5\).

### Step 2: Plug the known values into the equation

\[
10000 = 4000(1 + r)^5
\]

### Step 3: Solve for \(r\)

First, divide both sides by 4000:

\[
\frac{10000}{4000} = (1 + r)^5
\]

\[
2.5 = (1 + r)^5
\]

Now take the 5th root of both sides:

\[
(2.5)^{1/5} = 1 + r
\]

\[
1.2743 \approx 1 + r
\]

### Step 4: Isolate \(r\)

\[
r \approx 1.2743 - 1
\]

\[
r \approx 0.2743
\]

### Step 5: Convert to percentage

\[
r \approx 0.2743 \times 100 = 27.43\%
\]

Rounding to the nearest tenth:

\[
r \approx 27.4\%
\]

### Final Answer:
The rate of growth is approximately **27.4%**.

Would you like more details or clarifications on this process?

Here are 5 related questions:
1. What would the population be in 10 years given the same rate of growth?
2. How would the model change if the growth were logistic instead of exponential?
3. What is the doubling time for this population growth rate?
4. How can you calculate the continuous growth rate based on this data?
5. How would a decrease in the rate affect the future population?

**Tip**: To solve exponential growth problems, always first isolate the exponential term before taking roots or logarithms.

Suppose a fish population is currently 4000 and 5 years later the population is 10000. Use the explicit exponential model Pn=Po(1+r)^n, to find the rate of growth. Express the rate of growth as a percentage to the nearest tenth.

Learn how to calculate the rate of growth of a fish population that increases from 4000 to 10000 in 5 years using the exponential growth model. This problem involves key concepts in algebra and population modeling, suitable for high school students (Grades 9-12).

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