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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. 2,460 people
C. 2,584 people
D. 14,887 people


Sagot :

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