Answer :
To solve this problem, we are looking for the equation that models the salary over time with a [tex]$4\%$[/tex] annual raise. The raise results in a multiplier of [tex]$1.04$[/tex] each year, meaning the salary for year [tex]\(n\)[/tex] is the previous year's salary multiplied by [tex]$1.04$[/tex].
Let's break it down step by step:
1. Initial Salary: Finley's starting salary in the first year is [tex]$57,000.
2. Annual Increase Rate: Every year, the salary increases by $[/tex]4\%[tex]$. This means you multiply the salary of the previous year by $[/tex]1 + 0.04 = 1.04[tex]$.
3. Equation Structure: We need to find an equation that represents the salary after \(n\) years. The formula for the salary in year \(n\) is:
\[
f(n) = 57000 \times (1.04)^{n-1}
\]
Here's why:
- \(57000\) is the starting salary.
- \(1.04\) is the factor by which the salary is multiplied each year due to the $[/tex]4\%[tex]$ raise.
- \((n-1)\) is the exponent because the starting year is counted as \(n=1\) (the first year with no multiplier effect yet, hence \(n-1\)).
4. Correct Answer Choice: Look at the options provided:
- \(f(n)=57,000\left(0.04^{n-1}\right)\)
- \(f(n)=57,000\left(1.04^{\prime \prime}\right)\)
- \(f(n)=57,000\left(1.04^{n-1}\right)\)
- \(f(n)=57,000\left(0.96^{\prime \prime}\right)\)
The correct equation that fits our derived formula is:
\[
f(n) = 57,000 \times (1.04)^{n-1}
\]
Therefore, the correct answer is the third option: \(f(n)=57,000\left(1.04^{n-1}\right)\).
This equation correctly models the salary progression with a $[/tex]4\%$ annual raise starting from [tex]\(57,000\)[/tex] in the first year.
Let's break it down step by step:
1. Initial Salary: Finley's starting salary in the first year is [tex]$57,000.
2. Annual Increase Rate: Every year, the salary increases by $[/tex]4\%[tex]$. This means you multiply the salary of the previous year by $[/tex]1 + 0.04 = 1.04[tex]$.
3. Equation Structure: We need to find an equation that represents the salary after \(n\) years. The formula for the salary in year \(n\) is:
\[
f(n) = 57000 \times (1.04)^{n-1}
\]
Here's why:
- \(57000\) is the starting salary.
- \(1.04\) is the factor by which the salary is multiplied each year due to the $[/tex]4\%[tex]$ raise.
- \((n-1)\) is the exponent because the starting year is counted as \(n=1\) (the first year with no multiplier effect yet, hence \(n-1\)).
4. Correct Answer Choice: Look at the options provided:
- \(f(n)=57,000\left(0.04^{n-1}\right)\)
- \(f(n)=57,000\left(1.04^{\prime \prime}\right)\)
- \(f(n)=57,000\left(1.04^{n-1}\right)\)
- \(f(n)=57,000\left(0.96^{\prime \prime}\right)\)
The correct equation that fits our derived formula is:
\[
f(n) = 57,000 \times (1.04)^{n-1}
\]
Therefore, the correct answer is the third option: \(f(n)=57,000\left(1.04^{n-1}\right)\).
This equation correctly models the salary progression with a $[/tex]4\%$ annual raise starting from [tex]\(57,000\)[/tex] in the first year.
