Let's find the right elements to fill in the blanks and answer the question step by step.
### Step 1: Identify the variables and given information.
- Current population ([tex]\(N_0\)[/tex]) = 9,400 people
- Rate of decrease ([tex]\(r\)[/tex]) = 14.3% = 0.143
- Threshold population ([tex]\(P_{\text{threshold}}\)[/tex]) = 6,000 people
### Step 2: Write down the exponential decay formula.
The population after [tex]\(t\)[/tex] years, where the population decreases by a rate [tex]\(r\)[/tex] each year, can be expressed as:
[tex]\[ N(t) = N_0 \cdot (1 - r)^t \][/tex]
### Step 3: Create the inequality.
We need the population to be less than the threshold to trigger the tax status change:
[tex]\[ N_0 \cdot (1 - r)^t < P_{\text{threshold}} \][/tex]
Substitute the given values:
[tex]\[ 9400 \cdot (1 - 0.143)^t < 6000 \][/tex]
So, the inequality is:
[tex]\[ 9400 \cdot (0.857)^t < 6000 \][/tex]
### Step 4: Determine if the town's tax status will change within the next 3 years.
Using the same exponential decay formula to find the population after 3 years:
[tex]\[ N(3) = 9400 \cdot (0.857)^3 \][/tex]
From the given result, we know that after 3 years:
[tex]\[ N(3) = 5916.574254199999 \][/tex]
Since 5916.574254199999 is less than 6000, the population will indeed be below the threshold.
### Final step: Fill in the blanks and select the correct answer.
The inequality is:
[tex]\[ 9400 \ (0.857)^t < 6000 \][/tex]
Will the town's tax status change within the next 3 years? Yes.
So, the correct selections are:
- [tex]\(9400\)[/tex]
- [tex]\((0.857)\)[/tex]
- [tex]\(6000\)[/tex]
- Yes
