To determine after how many years, [tex]\( t \)[/tex], the town's tax status will change, we need to create an inequality based on exponential decay. The population is decreasing at an exponential rate of [tex]\( 14.3\% \)[/tex] per year, which means the population each year is [tex]\( 85.7\% \)[/tex] ([tex]\( 100\% - 14.3\% = 85.7\% \)[/tex]) of the previous year’s population.
Given:
- Initial population ([tex]\( P_0 \)[/tex]): [tex]\( 9,400 \)[/tex] people
- Decreasing rate: [tex]\( 14.3\% \)[/tex] or [tex]\( 0.143 \)[/tex] in decimal
- Target population: [tex]\( 6,000 \)[/tex] people
The population after [tex]\( t \)[/tex] years can be modeled using the exponential decay formula:
[tex]\[ P(t) = P_0 \cdot (1 - \text{decreasing rate})^t \][/tex]
Substituting the known values:
[tex]\[ P(t) = 9400 \cdot (1 - 0.143)^t \][/tex]
[tex]\[ P(t) = 9400 \cdot (0.857)^t \][/tex]
We want to find the inequality for when the population falls below 6,000 people:
[tex]\[ 9400 \cdot (0.857)^t < 6000 \][/tex]
This is the inequality that can be used to determine after how many years the town's tax status will change.
Now, we need to check if the town's tax status will change within the next 3 years.
To do this, let’s evaluate the population after 3 years using the previous formula:
[tex]\[ P(3) = 9400 \cdot (0.857)^3 \][/tex]
The population after 3 years is approximately [tex]\( 5916.57 \)[/tex] people.
Since [tex]\( 5916.57 \)[/tex] is less than [tex]\( 6000 \)[/tex], the town's tax status will indeed change within the next 3 years.
So, the correct answers to select are:
1. [tex]\( 9400 \cdot (0.857)^t \)[/tex]
2. <
3. 6000
And:
Yes, the town's tax status will change within the next 3 years.
