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.