Sure! Let's work through the problem step-by-step:
1. Understand the regression equation: The population [tex]\( y \)[/tex] is modeled by the equation [tex]\( y = 20,000 \left(1.03^x\right) \)[/tex], where [tex]\( y \)[/tex] represents the population, [tex]\( 20,000 \)[/tex] is the initial population, [tex]\( 1.03 \)[/tex] is the growth rate, and [tex]\( x \)[/tex] is the number of years.
2. Substitute the given years into the equation: We need to predict the population after 25 years, so let [tex]\( x = 25 \)[/tex].
3. Calculate the population:
[tex]\[
y = 20,000 \left(1.03^{25}\right)
\][/tex]
When we compute the value of [tex]\( 1.03^{25} \)[/tex], it results in approximately 2.093.
4. Multiply by the initial population:
[tex]\[
y = 20,000 \times 2.093
\][/tex]
This gives us:
[tex]\[
y \approx 41,875.55859308432
\][/tex]
5. Compare with the options given:
- A. 50,002
- B. 41,876
- C. 34,049
- D. 30,252
The closest value to 41,875.55859308432 is:
- B. 41,876
Therefore, the best prediction for the population in year 25 is:
B. 41,876
