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The population of a small town is decreasing exponentially at a rate of [tex]14.3\%[/tex] each year. The current population is 9,400 people. The town's tax status will change once the population is below 6,000 people.

Create an inequality that can be used to determine after how many years, [tex]t[/tex], the town's tax status will change, and use it to answer the question below.

1. [tex]\[9400 \cdot (1 - 0.143)^t \ \textless \ 6000\][/tex]

2. Will the town's tax status change within the next 3 years?

[tex]\square[/tex]

Sagot :

GinnyAnswer
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]
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