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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. 7,380 people
B. 2,584 people
C. 2,460 people
D. 14,887 people



Answer :

GinnyAnswer
Certainly! To determine the population of the community 3 years after it started being recorded, we need to use the given exponential function for population growth:

[tex]\[ p(x) = 2400 \cdot (1.025)^x \][/tex]

Here,
- [tex]\( x \)[/tex] is the number of years since the population started being recorded.
- [tex]\( 2400 \)[/tex] is the initial population.
- [tex]\( 1.025 \)[/tex] is the growth rate per year.

Since we want the population after 3 years, we set [tex]\( x = 3 \)[/tex]:

[tex]\[ p(3) = 2400 \cdot (1.025)^3 \][/tex]

First, we need to calculate [tex]\( (1.025)^3 \)[/tex]:
[tex]\[ (1.025)^3 \approx 1.077 \][/tex]

Now, multiply this result by the initial population of 2400:
[tex]\[ p(3) = 2400 \cdot 1.077 \][/tex]

This gives us:
[tex]\[ p(3) \approx 2584.5375 \][/tex]

Rounding to a whole number, the population after 3 years is approximately:
[tex]\[ 2584 \][/tex]

So, the correct answer is:
[tex]\[ \text{B. 2,584 people} \][/tex]

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