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.