Question
Use the information provided to answer Part A and Part B.
Sam is earning money by mowing lawns, while his friend Will is getting paid to tutor. Sam's money can be represented by the function M = 25w + 75, where w is the number of weeks since Sam began working, and M is the total amount of money possessed in dollars. The table below models Will's money and earnings.
Will's Earnings
Number of Weeks Total Money ($)
0 140
1 160
2 180
3 200
Select the options that correctly complete the statements.
Part A
Sam's rate of change is (answer) Will's rate of change.
Part B
Sam's initial amount of money possessed before his mowing job is (answer) Will's initial amount before tutoring.
Sam is earning money by mowing lawns, while his friend Will is getting paid to tutor. Sam's money can be represented by the function M = 25w + 75, where w is the number of weeks since Sam began working, and M is the total amount of money possessed in dollars. The table below models Will's money and earnings.
Will's Earnings
Number of Weeks Total Money ($)
0 140
1 160
2 180
3 200
Select the options that correctly complete the statements.
Part A
Sam's rate of change is (answer) Will's rate of change.
Part B
Sam's initial amount of money possessed before his mowing job is (answer) Will's initial amount before tutoring.
Answers
GPT-4o mini
To answer the questions, we first need to analyze the information provided about Sam and Will.
**Part A: Rate of Change**
1. **Sam's Rate of Change**: The function for Sam's money is given by \( M = 25w + 75 \). The coefficient of \( w \) (which is 25) represents Sam's rate of change in money per week. This means that for every week, Sam earns an additional $25.
2. **Will's Rate of Change**: From the table, we can find the change in Will's total money over the weeks:
- From week 0 to week 1: \( 160 - 140 = 20 \)
- From week 1 to week 2: \( 180 - 160 = 20 \)
- From week 2 to week 3: \( 200 - 180 = 20 \)
Thus, Will's rate of change is also $20 per week.
With that, we can conclude for **Part A**:
**Sam's rate of change is greater than Will's rate of change.**
**Part B: Initial Amount of Money**
1. **Sam's Initial Amount**: From the function \( M = 25w + 75 \), when \( w = 0 \) (the beginning), we have:
\( M = 25(0) + 75 = 75 \). Therefore, Sam's initial amount of money is $75.
2. **Will's Initial Amount**: According to the table, at week 0, Will has $140.
Now we can conclude for **Part B**:
**Sam's initial amount of money possessed before his mowing job is less than Will's initial amount before tutoring.**
To summarize:
- **Part A**: Sam's rate of change is greater than Will's rate of change.
- **Part B**: Sam's initial amount of money possessed before his mowing job is less than Will's initial amount before tutoring.
**Part A: Rate of Change**
1. **Sam's Rate of Change**: The function for Sam's money is given by \( M = 25w + 75 \). The coefficient of \( w \) (which is 25) represents Sam's rate of change in money per week. This means that for every week, Sam earns an additional $25.
2. **Will's Rate of Change**: From the table, we can find the change in Will's total money over the weeks:
- From week 0 to week 1: \( 160 - 140 = 20 \)
- From week 1 to week 2: \( 180 - 160 = 20 \)
- From week 2 to week 3: \( 200 - 180 = 20 \)
Thus, Will's rate of change is also $20 per week.
With that, we can conclude for **Part A**:
**Sam's rate of change is greater than Will's rate of change.**
**Part B: Initial Amount of Money**
1. **Sam's Initial Amount**: From the function \( M = 25w + 75 \), when \( w = 0 \) (the beginning), we have:
\( M = 25(0) + 75 = 75 \). Therefore, Sam's initial amount of money is $75.
2. **Will's Initial Amount**: According to the table, at week 0, Will has $140.
Now we can conclude for **Part B**:
**Sam's initial amount of money possessed before his mowing job is less than Will's initial amount before tutoring.**
To summarize:
- **Part A**: Sam's rate of change is greater than Will's rate of change.
- **Part B**: Sam's initial amount of money possessed before his mowing job is less than Will's initial amount before tutoring.