After years of maintaining a steady population of 32,000, the population of a town begins to grow exponentially. After 1 year and an increase of 8% per year, the population is 34,560. Which equation can be used to predict, *y*, the number of people living in the town after *x* years? (Round population values to the nearest whole number.)

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.

Learn how to predict population growth using an exponential growth model. Solve the problem of predicting the population of a town after years of steady growth, with a detailed step-by-step solution. Suitable for high school students studying exponential functions and percentage increase.

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