Sure, let's go through the steps to determine the population of the town after 12 years given an initial population of 18,000 and a constant annual growth rate of 2%.
### Step-by-Step Solution
1. Identify the Information Provided:
- Initial population ([tex]\( P_0 \)[/tex]): 18,000
- Annual growth rate ([tex]\( r \)[/tex]): 2% or 0.02
- Number of years ([tex]\( t \)[/tex]): 12
2. Population Growth Formula:
The general formula for population growth with continuous compounding is:
[tex]\[
P(t) = P_0 \times (1 + r)^t
\][/tex]
Here:
- [tex]\( P(t) \)[/tex] is the population after [tex]\( t \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the annual growth rate.
- [tex]\( t \)[/tex] is the number of years.
3. Substitute the Values into the Formula:
[tex]\[
P(12) = 18,000 \times (1 + 0.02)^{12}
\][/tex]
This simplifies to:
[tex]\[
P(12) = 18,000 \times (1.02)^{12}
\][/tex]
4. Calculate the Growth Factor:
First, calculate [tex]\( (1.02)^{12} \)[/tex]:
[tex]\[
(1.02)^{12} \approx 1.26824
\][/tex]
5. Multiply the Initial Population by the Growth Factor:
[tex]\[
P(12) = 18,000 \times 1.26824 \approx 22828.32
\][/tex]
6. Round to the Nearest Whole Number:
The population after rounding to the nearest whole number is:
[tex]\[
\boxed{22828}
\][/tex]
### Conclusion
The population of the town after 12 years, growing at a rate of 2% per year, will be approximately 22,828 people.
