Sure, let's break down the steps required to find the solution to this problem.
### Step 1: Understanding Exponential Decay
The population of the town decreases exponentially at a rate of 14.3% each year. This can be described by the exponential decay formula:
[tex]\[
P(t) = P_0 \cdot (1 - r)^t
\][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population after [tex]\( t \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the rate of decay.
- [tex]\( t \)[/tex] is the number of years.
### Step 2: Setting Up the Inequality
You're given the initial population, [tex]\( P_0 = 9400 \)[/tex], and the rate of decay, [tex]\( r = 0.143 \)[/tex]. You need to determine how many years, [tex]\( t \)[/tex], it will take for the population to fall below the threshold of 6000.
So the inequality you need is:
[tex]\[
9400 \cdot (1 - 0.143)^t < 6000
\][/tex]
### Step 3: Determine if the Tax Status Changes Within the Next 3 Years
Substituting [tex]\( t = 3 \)[/tex] into the exponential decay formula:
[tex]\[
P(3) = 9400 \cdot (1 - 0.143)^3
\][/tex]
The population after 3 years is 5916.574254199999. Since 5916.574254199999 is less than 6000, the town's tax status will indeed change within the next 3 years.
### Answer
The correct inequality and answer to the question will be set as follows:
1. Fill in the drop-down menus for the inequality:
[tex]\[
9400 \cdot (1 - 0.143)^t < 6000
\][/tex]
2. Answer whether the town's tax status will change within the next 3 years:
[tex]\[
\text{Yes}
\][/tex]
So, the finalized view should read:
```
( 9400 * ( 1 - 0.143 ) ^ t < 6000 )
```
```
Will the town's tax status change within the next 3 years? Yes
```
All these calculations confirm that the town's population will drop below 6000 in less than 3 years and thus the tax status will change.
