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 [tex][tex]$8 \%$[/tex][/tex] per year, the population is 34,560. Which equation can be used to predict [tex]y[/tex], the number of people living in the town after [tex]x[/tex] years? (Round population values to the nearest whole number.)

A. [tex]y=32,000(1.08)^x[/tex]

B. [tex]y=32,000(0.08)^x[/tex]

C. [tex]y=34,560(1.08)^x[/tex]

D. [tex]y=34,560(0.08)^x[/tex]



Answer :

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]