To determine which equation best models the population of carpenter ants over time, we need to analyze the given information and follow a logical approach. Let's proceed step-by-step:
1. Understand the Growth Rate:
- The population increases by 23% every 2 months. This can be expressed mathematically as a growth factor of 1.23 (since increasing by 23% means multiplying by 1.23).
2. Initial Population:
- At the beginning of 2000, the population is 100 ants.
3. Modeling Over Time:
- The goal is to find a general formula that models the population `A` after `n` years. Note that each year consists of 12 months.
4. Conversion of Time Units:
- Since the growth period given (2 months) is different from the period we want to express the growth (years), we need to convert between these units correctly.
5. Calculate the Number of 2-Month Periods in a Year:
- There are 12 months in a year.
- Hence, there are [tex]\( \frac{12}{2} = 6 \)[/tex] periods of 2 months in a year.
6. Determine the Exponential Growth Expression:
- If the population grows by a factor of 1.23 every 2 months, then in 1 year (which has 6 such periods), the population will grow by a factor of [tex]\( (1.23)^6 \)[/tex].
- For [tex]\( n \)[/tex] years, there would be [tex]\( 6n \)[/tex] such periods of 2 months.
- Therefore, the general expression for the population after [tex]\( n \)[/tex] years would be:
[tex]\[
A = 100 \cdot (1.23)^{6n}
\][/tex]
7. Matching with the Provided Options:
- Now we need to match this expression with the provided options:
- A) [tex]\( A = 100(1.23)^{\frac{n}{2}} \)[/tex]
- B) [tex]\( A = 100(1.23)^{2n} \)[/tex]
- C) [tex]\( A = 100(1.23)^{\frac{n}{6}} \)[/tex]
- D) [tex]\( A = 100(1.23)^{6n} \)[/tex]
8. Conclusion:
- The expression [tex]\( A = 100 \cdot (1.23)^{6n} \)[/tex] matches option D perfectly.
Thus, the correct equation that models the population of carpenter ants over time is:
[tex]\[ \boxed{D) \ A = 100(1.23)^{6n}} \][/tex]
