To address the problem, we will start by understanding the exponential decay formula used to model population decrease. The formula to calculate the future population, [tex]\( P(t) \)[/tex], after [tex]\( t \)[/tex] years is:
[tex]\[ P(t) = P_0 \times (1 - r)^t \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the annual decay rate,
- [tex]\( t \)[/tex] is the number of years.
Given the values:
- [tex]\( P_0 = 9400 \)[/tex] people,
- [tex]\( r = 0.143 \)[/tex] (which is 14.3%),
- [tex]\( t = 3 \)[/tex] years.
We will calculate the population after 1 year, 2 years, and 3 years.
1. Population after 1 year:
[tex]\[ P(1) = 9400 \times (1 - 0.143)^1 \][/tex]
[tex]\[ P(1) = 9400 \times 0.857 \][/tex]
[tex]\[ P(1) = 8055.8 \][/tex]
2. Population after 2 years:
[tex]\[ P(2) = 9400 \times (1 - 0.143)^2 \][/tex]
[tex]\[ P(2) = 9400 \times 0.857^2 \][/tex]
[tex]\[ P(2) = 9400 \times 0.7340329 \][/tex]
[tex]\[ P(2) = 6903.8206 \][/tex]
3. Population after 3 years:
[tex]\[ P(3) = 9400 \times (1 - 0.143)^3 \][/tex]
[tex]\[ P(3) = 9400 \times 0.857^3 \][/tex]
[tex]\[ P(3) = 9400 \times 0.6283838 \][/tex]
[tex]\[ P(3) = 5916.5743 \][/tex]
Next, we check if the population after 3 years is below the threshold of 6,000 people.
[tex]\[ 5916.5743 < 6000 \][/tex]
Since [tex]\( 5916.5743 \)[/tex] (population after 3 years) is less than 6000, the town's tax status will indeed change within the next 3 years.
Finally, we form the inequality to determine the number of years, [tex]\( t \)[/tex], at which the population will go below 6,000 people. The inequality is:
[tex]\[ 9400 \times (1 - 0.143)^t < 6000 \][/tex]
To answer the question:
1. The correct inequality: [tex]\( 9400 \times (1 - 0.143)^t < 6000 \)[/tex]
2. Will the town's tax status change within the next 3 years?: Yes
So, the completed answer in the drop-down menu format is:
[tex]\[
\begin{aligned}
& \text{\_\_\, \_\_, } \quad 9400 \times (1 - 0.143)^t < 6000
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
& \text{\_\_\, \_\_, } \quad \text{Yes}
\end{aligned}
\][/tex]
