To solve this problem, let's analyze the given information and determine which equation best models the exponential growth of the town's population.
### Step-by-Step Solution:
1. Initial Population and Growth Information:
- The initial population of the town is 32,000 people.
- The population grows at an exponential rate of 8% per year.
- After one year, the population is given as 34,560 people.
2. Understanding Exponential Growth:
Exponential growth can be modeled using the equation:
[tex]\[
y = P(1 + r)^x
\][/tex]
where:
- [tex]\(y\)[/tex] is the future population,
- [tex]\(P\)[/tex] is the initial population,
- [tex]\(r\)[/tex] is the growth rate,
- [tex]\(x\)[/tex] is the number of years.
3. Substituting Values:
- Initial population, [tex]\(P\)[/tex] = 32,000.
- Growth rate, [tex]\(r\)[/tex] = 8% = 0.08.
4. Equation for the Population After [tex]\(x\)[/tex] Years:
Substituting [tex]\(P\)[/tex] and [tex]\(r\)[/tex] into the exponential growth equation, we get:
[tex]\[
y = 32,000(1 + 0.08)^x = 32,000(1.08)^x
\][/tex]
5. Verification with Given Data:
To verify, we can check if this equation fits the population after one year ([tex]\(x = 1\)[/tex]):
[tex]\[
y = 32,000(1.08)^1 = 32,000 \times 1.08 = 34,560
\][/tex]
This matches the provided population after one year, confirming that our equation is correct.
6. Choosing the Correct Equation from the Options:
The provided options are:
- [tex]\(y = 32,000(1.08)^x\)[/tex]
- [tex]\(y = 32,000(0.08)^x\)[/tex]
- [tex]\(y = 34,560(1.08)^x\)[/tex]
- [tex]\(y = 34,560(0.08)^x\)[/tex]
Among these options, the correct equation is:
[tex]\[
y = 32,000(1.08)^x
\][/tex]
### Conclusion:
The correct equation that predicts the number of people living in the town after [tex]\(x\)[/tex] years, considering the initial population and the given growth rate, is:
[tex]\[ y = 32,000(1.08)^x \][/tex]
Thus, the answer is:
[tex]\[ y = 32,000(1.08)^x \][/tex]
This equation correctly models the exponential growth of the town's population over time.
