The population of the village of Brigethville, Florida has been observed to grow exponentially. The table shows the population for a six year period beginning in 2016.

Year	2016	2017	2018	2019	2020	2021
Population	790	894	1054	1267	1477	1884


Find an exponential equation that best fits the data above. 

To find an exponential equation that best fits the data, we need to model the population \( P(t) \) of Brigethville as a function of time \( t \). The general form of an exponential growth equation is:

\[
P(t) = P_0 \cdot e^{kt}
\]

Where:
- \( P(t) \) is the population at time \( t \) (in years since 2016),
- \( P_0 \) is the initial population (at \( t = 0 \), i.e., the year 2016),
- \( k \) is the growth rate,
- \( t \) is the number of years since 2016.

### Steps to find the equation:

1. **Identify initial values:**
   - From the table, the initial population in 2016 is \( P_0 = 790 \).
   - \( t = 0 \) corresponds to 2016, so for 2017, \( t = 1 \), for 2018, \( t = 2 \), etc.

2. **Choose two points to solve for \( k \):**
   Let's pick the population values for 2016 (\( t = 0 \)) and 2021 (\( t = 5 \)) for simplicity.
   - In 2016, \( P(0) = 790 \),
   - In 2021, \( P(5) = 1884 \).

3. **Set up the equation for 2021:**
   \[
   1884 = 790 \cdot e^{5k}
   \]

4. **Solve for \( k \):**
   Divide both sides by 790:
   \[
   \frac{1884}{790} = e^{5k}
   \]

   \[
   2.3848 = e^{5k}
   \]

   Take the natural logarithm (ln) of both sides:
   \[
   \ln(2.3848) = 5k
   \]

   \[
   k = \frac{\ln(2.3848)}{5} = \frac{0.8689}{5} \approx 0.1738
   \]

5. **Write the exponential equation:**
   Now that we know \( P_0 = 790 \) and \( k \approx 0.1738 \), the exponential equation is:
   \[
   P(t) = 790 \cdot e^{0.1738t}
   \]

This equation models the population growth of Brigethville from 2016 onward.

### Would you like me to verify the fit of this equation using other data points or explain the calculation steps further?

### Related Questions:
1. How can you predict the population for a future year, say 2025, using this equation?
2. What does the growth rate \( k \) represent in real-world terms?
3. How would the equation change if the population grew at a slower or faster rate?
4. Can you calculate the population doubling time based on the growth rate?
5. How does the exponential growth model compare to a linear growth model for this data?

### Tip:
When modeling data with exponential growth, using logarithmic transformations can make it easier to find the growth rate.

The population of the village of Brigethville, Florida has been observed to grow exponentially. The table shows the population for a six-year period beginning in 2016.

Year: 2016, 2017, 2018, 2019, 2020, 2021
Population: 790, 894, 1054, 1267, 1477, 1884

Find an exponential equation that best fits the data above.

Discover how to model the population growth of Brigethville, Florida, using an exponential growth equation based on data from 2016 to 2021. This tutorial explains the key steps, including identifying the initial population and solving for the growth rate. Suitable for students in Grades 9-12, this problem focuses on exponential functions and algebraic modeling.

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