Let's solve the problem step by step.
1. Understanding the Problem:
- We are given an exponential function that models the population, [tex]\( p(x) = 2,400 \times (1.025)^x \)[/tex], where [tex]\( x \)[/tex] is the number of years.
- We need to find the population after 3 years (i.e., when [tex]\( x = 3 \)[/tex]).
2. Substitute [tex]\( x = 3 \)[/tex] into the Exponential Function:
- The function becomes:
[tex]\[
p(3) = 2,400 \times (1.025)^3
\][/tex]
3. Calculate [tex]\( (1.025)^3 \)[/tex]:
- Using the exponential base of 1.025 raised to the power of 3,
[tex]\[
(1.025)^3 \approx 1.0775625
\][/tex]
(This value can be obtained using a calculator or exponentiation rules)
4. Multiply by the Initial Population:
- Now, we need to multiply this result by the initial population of 2,400:
[tex]\[
p(3) \approx 2,400 \times 1.0775625 = 2,584.5375
\][/tex]
5. Rounding to the Nearest Whole Number:
- Since population count should be a whole number (you cannot have a fraction of a person), we round 2,584.5375 to the nearest whole number:
[tex]\[
\approx 2,585
\][/tex]
6. Conclusion:
- The approximate population 3 years after recording started is 2,585 people.
- Among the given options (A. 14,887 people, B. 2,460 people, C. 7,380 people, D. 2,584 people), the closest and most accurate answer based on our calculation is:
D. 2,584 people.
Thus, the correct answer is D. 2,584 people.
