To determine the approximate population 3 years after the population started being recorded, we need to use the given exponential model for the population growth, which is:
[tex]\[ p(x) = 2,400 \times (1.025)^x \][/tex]
In this model:
- [tex]\( p(x) \)[/tex] represents the population after [tex]\( x \)[/tex] years.
- [tex]\( 2,400 \)[/tex] is the initial population.
- [tex]\( 1.025 \)[/tex] is the growth rate.
- [tex]\( x \)[/tex] is the number of years since the population started being recorded.
Since we want to know the population after 3 years, we plug [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[ p(3) = 2,400 \times (1.025)^3 \][/tex]
Now let's interpret this step-by-step:
1. Starting with the initial population of 2,400.
2. The growth rate is 1.025, which means the population increases by 2.5% each year.
3. We need to calculate the population after 3 years by raising the growth rate (1.025) to the power of 3 and then multiplying it by the initial population (2,400).
Performing this calculation:
[tex]\[ p(3) = 2,400 \times (1.025)^3 \approx 2,584.54 \][/tex]
Upon rounding to the nearest whole number, we obtain approximately:
[tex]\[ p(3) \approx 2,584 \][/tex]
Therefore, the closest answer to the calculated population after 3 years is:
C. 2,584 people
