Respuesta :
Using an exponential function, the inequality is given as follows:
[tex]9400(0.857)^t < 6000[/tex]
The solution is t > 2.9, hence the tax status will change within the next 3 years.
What is an exponential function?
A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
For this problem, the parameters are given as follows:
A(0) = 9400, r = 0.143.
The population after t years is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
[tex]A(t) = 9400(1 - 0.143)^t[/tex]
[tex]A(t) = 9400(0.857)^t[/tex]
The tax status will change when:
[tex]A(t) < 6000[/tex]
Hence the inequality is:
[tex]9400(0.857)^t < 6000[/tex]
Then:
[tex](0.857)^t < \frac{6000}{9400}[/tex]
[tex]\log{(0.857)^t} < \log{\left(\frac{6000}{9400}\right)}[/tex]
[tex]t\log{0.857} < \log{\left(\frac{6000}{9400}\right)}[/tex]
Since both logs are negative:
[tex]t > \frac{\log{\left(\frac{6000}{9400}\right)}}{\log{0.857}}[/tex]
t > 2.9.
The solution is t > 2.9, hence the tax status will change within the next 3 years.
More can be learned about exponential functions at https://brainly.com/question/25537936
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