Find an exponential equation that best fits the data above. The equation has form P t

To find the exponential equation that best fits the data of population growth, we aim for a model of the form:

$P(t) = a \cdot b^t$

Where:

• $P(t)$ is the population at year $t$,
• $t$ is the number of years since 1990, meaning $t = 1$ for 1991, $t = 2$ for 1992, and so on,
• $a$ and $b$ are constants to be determined through curve-fitting.

Step-by-step approach:

1. Transform the data: Set up the $t$ values as years since 1990:

   • $t = 1$ for 1991: Population = 1068
   • $t = 2$ for 1992: Population = 1804
   • $t = 3$ for 1993: Population = 3176
   • $t = 4$ for 1994: Population = 5582
   • $t = 5$ for 1995: Population = 9253
   • $t = 6$ for 1996: Population = 15177

2. Use curve-fitting methods: Applying curve fitting on the given data to determine $a$ and $b$, I will calculate this using least-squares regression for exponential growth.

Let me compute that for you.The exponential equation that best fits the given population data is:

$P(t) = 697.69 \cdot (1.6722)^t$

Where:

• $t$ is the number of years since 1990,
• $a \approx 697.69$,
• $b \approx 1.6722$.

This model suggests that the population grows by approximately 67.22% each year since 1990.

Would you like more details or have any questions?

Here are some related questions:

1. How accurate is this exponential model over a longer period?
2. Can this model be used to predict future population values?
3. How can you check if this exponential fit is the best for your data?
4. What happens to the population according to this model after 10 years?
5. Can we model this data with a polynomial function instead?

Tip: When fitting curves to data, always consider the residuals (the differences between observed and predicted values) to assess how well the model fits the data.