Let's work through the problem step-by-step to find the correct equation that predicts the population of the town after [tex]\( x \)[/tex] years.
1. Understand the Initial Conditions:
- Initial population: [tex]\( 32,000 \)[/tex]
- Growth rate: [tex]\( 8\% \)[/tex] per year, or a multiplier of [tex]\( 1.08 \)[/tex]
2. Construct the General Exponential Growth Formula:
- Population after [tex]\( x \)[/tex] years is given by [tex]\( y = P(1 + r)^x \)[/tex]
- Here, [tex]\( P \)[/tex] is the initial population and [tex]\( r \)[/tex] is the annual growth rate.
3. Substitute the Initial Population and Growth Rate into the Formula:
- Initial population [tex]\( P = 32,000 \)[/tex]
- Growth rate [tex]\( r = 0.08 \)[/tex]
- Therefore, the equation becomes:
[tex]\[
y = 32,000 \times (1.08)^x
\][/tex]
4. Verify the Equation with Given Data:
- Population after 1 year can be found by substituting [tex]\( x = 1 \)[/tex] into the equation:
[tex]\[
y = 32,000 \times (1.08)^1 = 32,000 \times 1.08 = 34,560
\][/tex]
- This matches the given population of 34,560 after 1 year.
With all steps verifying the growth equation, we can confidently conclude:
The correct equation that predicts the number of people living in the town after [tex]\( x \)[/tex] years is:
[tex]\[
y = 32,000 \times (1.08)^x
\][/tex]
