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The question asks to find an exponential equation that fits the population data for Rachelleville, Kentucky, based on the model \( P_t = ab^t \), where \( t \) is the number of years since 2015, and \( P_t \) is the population. 

### Given dataThe question asks to find an exponential equation that fits the population data for Rachelleville, Kentucky, based on the model \( P_t = ab^t \), where \( t \) is the number of years since 2015, and \( P_t \) is the population. 

### Given 
- Year 2016: Population = 771
- Year 2017: Population = 1022
- Year 2018: Population = 1253
- Year 2019: Population = 1579
- Year 2020: Population = 2203
- Year 2021: Population = 2864

We need to determine \( a \) and \( b \) using the curve-fitting process for this exponential model.

### Steps:
1. The base year is 2015, so let \( t = 0 \) correspond to the year 2015.
2. For 2016, \( t = 1 \); for 2017, \( t = 2 \); for 2018, \( t = 3 \), and so on.
3. The exponential equation is \( P_t = ab^t \), where \( a \) is the initial population in 2015 (before 2016) and \( b \) is the growth rate.

#### Step 1: Estimation of Parameters
We can apply logarithmic transformations and curve-fitting to estimate the values of \( a \) and \( b \) through regression or logarithmic methods.

Let me compute these values for you.

The exponential equation that best fits the given population data is:

\[
P_t = 567.56 \times (1.3086)^t
\]

Where:
- \( a = 567.56 \) (the estimated population in 2015),
- \( b = 1.3086 \) (the annual growth factor).

### Prediction for 2025:
For \( t = 2025 - 2015 = 10 \) years after 2015, we can predict the population in 2025.

Let me compute this prediction for you.The predicted population of Rachelleville, Kentucky, in the year 2025 is approximately **8,356** people, based on the exponential growth trend.

Would you like any further details or explanations?

### Related questions:
1. How would the growth rate change if the population increases faster than expected?
2. Can this model be applied to any population over a long period of time?
3. What happens if the population growth starts to slow down or stabilize?
4. How accurate are exponential models in real-life demographic studies?
5. How could external factors (such as migration) affect this exponential model?

**Tip:** Exponential models are best used for short-term predictions as real-world constraints (like resources) may slow growth in the long run.

Find an exponential equation that best fits the population data for Rachelleville, Kentucky (2016–2021). Predict the population in 2025.

Learn how to model the population growth of Rachelleville, Kentucky using an exponential growth equation. This example covers key concepts in exponential functions, including curve fitting and regression analysis. Predict the village's population in 2025 based on past trends (2016–2021). Suitable for students in Grades 9-12 or college.

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