To determine which equation best represents the model's prediction for the population of Springfield given the initial values and growth rate, let's evaluate each option one by one.
Given values:
- Initial population (in 2005): [tex]\( 15,000 \)[/tex]
- Annual growth rate: [tex]\( 4\% \)[/tex]
- Number of years after 2005 for evaluation: [tex]\( x = 5 \)[/tex]
The model predicts that the population increased by [tex]\( 4\% \)[/tex] of the previous year's population each year. This implies the population each year is multiplied by [tex]\( 1.04 \)[/tex] (which is [tex]\( 100\% + 4\% = 104\% \)[/tex]).
Let's analyze each equation:
Option (A): [tex]\( p = 0.96(15,000)^x \)[/tex]
We calculate this for [tex]\( x = 5 \)[/tex]:
[tex]\[ p_A = 0.96 \times (15,000)^5 \][/tex]
[tex]\[ p_A \approx 7.29 \times 10^{20} \][/tex]
Option (B): [tex]\( p = 1.04(15,000)^x \)[/tex]
We calculate this for [tex]\( x = 5 \)[/tex]:
[tex]\[ p_B = 1.04 \times (15,000)^5 \][/tex]
[tex]\[ p_B \approx 7.8975 \times 10^{20} \][/tex]
Option (C): [tex]\( p = 15,000(0.96)^x \)[/tex]
We calculate this for [tex]\( x = 5 \)[/tex]:
[tex]\[ p_C = 15,000 \times (0.96)^5 \][/tex]
[tex]\[ p_C \approx 12,230.59 \][/tex]
Option (D): [tex]\( p = 15,000(1.04)^x \)[/tex]
We calculate this for [tex]\( x = 5 \)[/tex]:
[tex]\[ p_D = 15,000 \times (1.04)^5 \][/tex]
[tex]\[ p_D \approx 18,249.79 \][/tex]
Now, let’s compare these results with the given population increase model's prediction:
- The population starting at [tex]\( 15,000 \)[/tex] should grow every year by [tex]\( 4\% \)[/tex]. This means if [tex]\( x = 5 \)[/tex], the population should be around [tex]\( 18,249.79 \)[/tex].
From these results, it is clear that option (D) [tex]\( p = 15,000(1.04)^x \)[/tex] best represents the model's prediction for the population growth in Springfield, as it accurately calculates the population increase by [tex]\( 4\% \)[/tex] each year for five years.
