To determine the population of the community 3 years after the population started being recorded, we begin with the exponential function given:
[tex]\[ p(x) = 2,400 (1.025)^x \][/tex]
Here, [tex]\(x\)[/tex] represents the number of years since the population started being recorded. We are tasked with finding the population for [tex]\(x = 3\)[/tex].
Substitute 3 into the function:
[tex]\[ p(3) = 2,400 \times (1.025)^3 \][/tex]
To proceed, let's break it down step by step.
1. Identify the initial population, growth rate, and years:
- Initial population, [tex]\(p_0\)[/tex]: 2,400 people
- Growth rate: 1.025
- Number of years, [tex]\(x\)[/tex]: 3
2. Compute the exponentiation:
- First, calculate [tex]\((1.025)^3\)[/tex]. When calculated, this is approximately [tex]\(1.0776\)[/tex].
3. Multiply by the initial population:
- Multiply the initial population by the result from the exponentiation step:
[tex]\[
p(3) \approx 2,400 \times 1.0776 \approx 2,584.5375
\][/tex]
4. Round to the nearest whole number if necessary:
- The approximate population is around 2,584 people.
Therefore, the approximate population 3 years after the population started being recorded is:
D. 2,584 people
