Select the correct answer.

The population of a community, [tex] p(x) [/tex], is modeled by this exponential function, where [tex] x [/tex] represents the number of years since the population started being recorded.

[tex]\[ p(x) = 2,400(1.025)^x \][/tex]

What is the approximate population 3 years after the population started being recorded?

A. 14,887 people

B. 2,460 people

C. 7,380 people

D. 2,584 people



Answer :

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.