To determine the equation that models the exponential population growth, we can use the general form of an exponential growth equation:

$y = a \cdot (1 + r)^x$

Where:

• $y$ is the population after $x$ years.
• $a$ is the initial population.
• $r$ is the growth rate (expressed as a decimal).
• $x$ is the number of years.

Given dataTo determine the equation that models the exponential population growth, we can use the general form of an exponential growth equation:

$y = a \cdot (1 + r)^x$

Where:

• $y$ is the population after $x$ years.
• $a$ is the initial population.
• $r$ is the growth rate (expressed as a decimal).
• $x$ is the number of years.

Given

1. Initial population, $a = 32,000$.
2. Growth rate, $r = 8\% = 0.08$.

Step-by-step derivation of the equation:

1. Plugging in the known values into the formula:

$y = 32000 \cdot (1 + 0.08)^x$

$y = 32000 \cdot (1.08)^x$

2. Verifying with given data point:

   To confirm this equation, we can check if it gives the population after 1 year as specified.

   $y = 32000 \cdot (1.08)^1$

   $y = 32000 \cdot 1.08$

   $y = 34560$

This matches the given information that the population is 34,560 after 1 year.

Thus, the equation that predicts the population $y$ after $x$ years is:

$\boxed{y = 32000 \cdot (1.08)^x}$

Would you like more details or have any questions?

Here are some related questions to explore:

1. What will the population be after 5 years using this model?
2. How long will it take for the population to double?
3. What happens to the population if the growth rate changes to 10%?
4. How does an exponential growth model compare to a linear growth model for population predictions?
5. What would the equation look like if the initial population was 50,000 instead of 32,000?

Tip: When dealing with exponential growth, the base of the exponent (in this case, 1.08) represents the growth factor per time unit.