https://file.aiquickdraw.com/mathbot/file/1728102326144i4cvi1oe.jpg

To solve this problem, we are tasked with finding an exponential equation in the form:

\[
P_t = a \cdot b^t
\]

Where:
- \( P_t \) is the population after \( t \) years (since 2015).
- \( a \) is the initial population (at \( t = 0 \)).
- \( b \) is the growth factor.
- \( t \) is the time in years since 2015.

### Given Data:
- For the year 2016 (\( t = 1 \)): Population = 771
- For the year 2017 (\( t = 2 \)): Population = 1022
- For the year 2018 (\( t = 3 \)): Population = 1253
- For the year 2019 (\( t = 4 \)): Population = 1579
- For the year 2020 (\( t = 5 \)): Population = 2203
- For the year 2021 (\( t = 6 \)): Population = 2864

### Step 1: Calculate \( a \) and \( b \)
- \( a \) represents the population at \( t = 0 \), which would correspond to the population in 2015. However, this isn't given directly in the table, so we can use curve fitting or regression to estimate both \( a \) and \( b \).

From the image, you have already calculated \( b = 1.2968136483244 \).

Now, to determine \( a \), we can use one of the given points (say, for \( t = 1 \) when \( P_1 = 771 \)):

\[
P_1 = a \cdot b^1
\]
\[
771 = a \cdot 1.2968136483244
\]
\[
a = \frac{771}{1.2968136483244} \approx 594.66
\]

So, \( a \approx 594.66 \).

### Step 2: Prediction for the year 2025
To predict the population in 2025, \( t = 2025 - 2015 = 10 \). Using the equation:

\[
P_{10} = a \cdot b^{10}
\]
\[
P_{10} = 594.66 \cdot (1.2968136483244)^{10}
\]

Calculating \( b^{10} \):

\[
b^{10} = (1.2968136483244)^{10} \approx 9.536
\]

Now, multiply this by \( a \):

\[
P_{10} \approx 594.66 \cdot 9.536 \approx 5670.44
\]

So, the predicted population in 2025 is approximately **5670** people (rounded to the nearest whole number).

---

Would you like further clarification or details? Here are 5 related questions you might be interested in:

1. How is the exponential growth model different from linear growth?
2. How would the equation change if the population growth slowed down in the future?
3. How does the value of \( b \) influence the rate of growth in the population?
4. What real-world factors might affect whether this exponential model holds true for future years?
5. Can this model be used to predict past population data (before 2016)?

**Tip:** In exponential models, the base \( b \) tells you the growth rate: if \( b > 1 \), there's growth; if \( 0 < b < 1 \), there's decay.

Find an exponential equation that best fits the data for the population of Rachelleville, Kentucky from 2016 to 2021, and predict the population for 2025 if the trend continues.

Solve the problem of predicting Rachelleville's population in 2025 based on exponential growth data from 2016 to 2021. Learn how to use the exponential growth formula P_t = a * b^t for accurate predictions.

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