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.