Let's address each part of the problem step by step.
### 1. Create an equation for each situation
For Job A:
- Initial salary: \[tex]$75,000
- Annual raise: \$[/tex]5,000 after the first year
The salary for Job A after [tex]\( n \)[/tex] years can be represented as:
[tex]\[ S_A(n) = 75000 + 5000n \][/tex]
For Job B:
- Initial salary: \[tex]$55,000
- Annual raise: 8% of the previous year's salary
The salary for Job B after \( n \) years can be expressed as:
\[ S_B(n) = 55000 \times (1 + 0.08)^n \]
### 2. Use each equation to complete the table
We will calculate the salaries for both jobs over the first 10 years.
| Year | Job A Salary (\$[/tex]) | Job B Salary (\[tex]$) |
|------|--------------------|----------------------------|
| 1 | 80,000 | 59,400 |
| 2 | 85,000 | 64,152 |
| 3 | 90,000 | 69,284.16 |
| 4 | 95,000 | 74,826.89 |
| 5 | 100,000 | 80,813.04 |
| 6 | 105,000 | 87,278.09 |
| 7 | 110,000 | 94,260.33 |
| 8 | 115,000 | 101,801.16 |
| 9 | 120,000 | 109,945.25 |
| 10 | 125,000 | 118,740.87 |
These values are computed using the given equations.
### 3. When will Job B earn more than Job A?
To determine when Job B's salary will exceed Job A's salary, we need to compare the salaries year by year.
From the table:
- In year 1, Job A: \$[/tex]80,000, Job B: \[tex]$59,400
- In year 2, Job A: \$[/tex]85,000, Job B: \[tex]$64,152
- In year 3, Job A: \$[/tex]90,000, Job B: \[tex]$69,284.16
- In year 4, Job A: \$[/tex]95,000, Job B: \[tex]$74,826.89
- In year 5, Job A: \$[/tex]100,000, Job B: \[tex]$80,813.04
- In year 6, Job A: \$[/tex]105,000, Job B: \[tex]$87,278.09
- In year 7, Job A: \$[/tex]110,000, Job B: \[tex]$94,260.33
- In year 8, Job A: \$[/tex]115,000, Job B: \[tex]$101,801.16
- In year 9, Job A: \$[/tex]120,000, Job B: \[tex]$109,945.25
- In year 10, Job A: \$[/tex]125,000, Job B: \[tex]$118,740.87
We observe that throughout this 10-year period, Job B's salary never exceeds Job A's salary. Job B's salary continues to increase but does not surpass Job A's salary within these 10 years.
### 4. Which job would you choose and why?
Given the observations from the table and the comparison over the 10 years:
- Job A starts at \$[/tex]75,000 and increases by \[tex]$5,000 each year.
- Job B starts at \$[/tex]55,000 with an 8% annual raise but does not exceed Job A's salary within the first 10 years.
Based on this data, I would choose Job A because Job A consistently offers a higher salary compared to Job B over the provided 10-year period. This higher consistent salary from Job A could lead to better financial stability and higher overall earnings in this timeframe.
