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Sagot :
Let's determine Carlos's salary for each job offer after \( t \) years.
### Job Offer 1: Fixed $4380 Raise Per Year
For this job, Carlos receives a fixed annual raise of $4380. The formula to calculate the salary after \( t \) years is:
[tex]\[ \text{Salary} = \text{Starting Salary} + (\text{Fixed Annual Raise} \times t) \][/tex]
Given:
- Starting salary \( = \$73000 \)
- Fixed annual raise \( = \$4380 \)
Let's calculate the salary for \( t = 1 \), \( t = 5 \), \( t = 10 \), and \( t = 15 \) years:
1. After 1 year:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 1) = \$77380
\][/tex]
2. After 5 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 5) = \$94900
\][/tex]
3. After 10 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 10) = \$116800
\][/tex]
4. After 15 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 15) = \$138700
\][/tex]
### Job Offer 2: 5% Raise Per Year
For this job, Carlos receives an annual raise of 5% of his current salary. The formula to calculate the salary after \( t \) years is:
[tex]\[ \text{Salary} = \text{Starting Salary} \times (1 + \text{Percent Annual Raise})^t \][/tex]
Given:
- Starting salary \( = \$73000 \)
- Percent annual raise \( = 0.05 \)
Let's calculate the salary for \( t = 1 \), \( t = 5 \), \( t = 10 \), and \( t = 15 \) years:
1. After 1 year:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^1 = \$[/tex]76650
\][/tex]
2. After 5 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^5 = \$[/tex]93169
\][/tex]
3. After 10 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^{10} = \$[/tex]118909
\][/tex]
4. After 15 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^{15} = \$[/tex]151762
\][/tex]
### Summary Tables
Here are the completed tables for both job offers:
#### Salary with $4380 Raise Per Year
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$t$[/tex] years & 1 & 5 & 10 & 15 \\
\hline
\begin{tabular}{c} Salary with \\ [tex]$4380$[/tex] raise \\ per year \end{tabular} & [tex]$77380 & $[/tex]94900 & [tex]$116800 & $[/tex]138700 \\
\hline
\end{tabular}
#### Salary with 5% Raise Per Year
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$t$[/tex] years & 1 & 5 & 10 & 15 \\
\hline
\begin{tabular}{c} Salary with \\ [tex]$5\%$[/tex] raise \\ per year \end{tabular} & [tex]$76650 & $[/tex]93169 & [tex]$118909 & $[/tex]151762 \\
\hline
\end{tabular}
This should give Carlos a clear understanding of his salary progression under each job offer over 15 years.
### Job Offer 1: Fixed $4380 Raise Per Year
For this job, Carlos receives a fixed annual raise of $4380. The formula to calculate the salary after \( t \) years is:
[tex]\[ \text{Salary} = \text{Starting Salary} + (\text{Fixed Annual Raise} \times t) \][/tex]
Given:
- Starting salary \( = \$73000 \)
- Fixed annual raise \( = \$4380 \)
Let's calculate the salary for \( t = 1 \), \( t = 5 \), \( t = 10 \), and \( t = 15 \) years:
1. After 1 year:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 1) = \$77380
\][/tex]
2. After 5 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 5) = \$94900
\][/tex]
3. After 10 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 10) = \$116800
\][/tex]
4. After 15 years:
[tex]\[ \text{Salary} = \[tex]$73000 + (\$[/tex]4380 \times 15) = \$138700
\][/tex]
### Job Offer 2: 5% Raise Per Year
For this job, Carlos receives an annual raise of 5% of his current salary. The formula to calculate the salary after \( t \) years is:
[tex]\[ \text{Salary} = \text{Starting Salary} \times (1 + \text{Percent Annual Raise})^t \][/tex]
Given:
- Starting salary \( = \$73000 \)
- Percent annual raise \( = 0.05 \)
Let's calculate the salary for \( t = 1 \), \( t = 5 \), \( t = 10 \), and \( t = 15 \) years:
1. After 1 year:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^1 = \$[/tex]76650
\][/tex]
2. After 5 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^5 = \$[/tex]93169
\][/tex]
3. After 10 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^{10} = \$[/tex]118909
\][/tex]
4. After 15 years:
[tex]\[ \text{Salary} = \[tex]$73000 \times (1 + 0.05)^{15} = \$[/tex]151762
\][/tex]
### Summary Tables
Here are the completed tables for both job offers:
#### Salary with $4380 Raise Per Year
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$t$[/tex] years & 1 & 5 & 10 & 15 \\
\hline
\begin{tabular}{c} Salary with \\ [tex]$4380$[/tex] raise \\ per year \end{tabular} & [tex]$77380 & $[/tex]94900 & [tex]$116800 & $[/tex]138700 \\
\hline
\end{tabular}
#### Salary with 5% Raise Per Year
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$t$[/tex] years & 1 & 5 & 10 & 15 \\
\hline
\begin{tabular}{c} Salary with \\ [tex]$5\%$[/tex] raise \\ per year \end{tabular} & [tex]$76650 & $[/tex]93169 & [tex]$118909 & $[/tex]151762 \\
\hline
\end{tabular}
This should give Carlos a clear understanding of his salary progression under each job offer over 15 years.
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