To determine the correct equation for predicting the population [tex]\( y \)[/tex] of the town after [tex]\( x \)[/tex] years, let's analyze the details given:
1. Initial Population:
The initial population of the town was 32,000 people.
2. Population Growth:
The population grows at a rate of 8\% per year.
3. Population After 1 Year:
After 1 year, the population grows to 34,560 people.
We need to find an equation that represents the population [tex]\( y \)[/tex] after [tex]\( x \)[/tex] years.
Here's how we determine the appropriate equation:
### Step-by-Step Analysis:
1. Growth Rate Converted to Decimal:
The growth rate of 8\% per year can be converted to decimal form by dividing by 100:
[tex]\[
8\% = \frac{8}{100} = 0.08
\][/tex]
2. Growth Factor:
The growth factor per year is calculated as:
[tex]\[
1 + 0.08 = 1.08
\][/tex]
3. Exponentiation for Growth Over [tex]\( x \)[/tex] Years:
Since the population grows exponentially, we use the exponential growth formula:
[tex]\[
y = \text{initial population} \times (\text{growth factor})^x
\][/tex]
4. Given Information:
- Initial population: 32,000
- Growth factor: 1.08
5. Equation for Predicting Population:
Substituting the initial population and the growth factor into the exponential growth formula, we get:
[tex]\[
y = 32,000 \times (1.08)^x
\][/tex]
### Verifying the Options:
Now, let's compare this result with the given options:
- [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]
Our derived equation matches the first option:
[tex]\[
y = 32,000(1.08)^x
\][/tex]
Therefore, the equation that can be used to predict the number of people living in the town after [tex]\( x \)[/tex] years, rounding population values to the nearest whole number, is:
[tex]\[
\boxed{y = 32,000(1.08)^x}
\][/tex]
