To solve this problem, we need to find the population of the community 3 years after it started being recorded, given the exponential function [tex]\( p(x) = 2400 \times (1.025)^x \)[/tex]. Here, [tex]\( x \)[/tex] represents the number of years, and we need to determine the value of [tex]\( p \)[/tex] when [tex]\( x = 3 \)[/tex].
Follow these steps:
1. Identify the initial population and the growth rate:
- Initial population [tex]\( P_0 = 2400 \)[/tex]
- Growth rate [tex]\( r = 1.025 \)[/tex]
2. Plug in the value of [tex]\( x \)[/tex] into the exponential function:
Since [tex]\( x = 3 \)[/tex], the function becomes:
[tex]\[
p(3) = 2400 \times (1.025)^3
\][/tex]
3. Calculate the value of the exponent:
Compute [tex]\( (1.025)^3 \)[/tex]:
[tex]\[
(1.025)^3 \approx 1.077
\][/tex]
4. Multiply the initial population by the computed exponential value:
So,
[tex]\[
p(3) = 2400 \times 1.077 \approx 2584.54
\][/tex]
5. Round to the nearest whole number:
[tex]\[
\approx 2585
\][/tex]
Hence, the approximate population 3 years after the population started being recorded is [tex]\( 2585 \)[/tex] people. However, since we are given options and comparing our calculated value to the closest available option, we see that the closest option is:
A. 2,584 people.
