Use the information provided to answer Part A and Part B.

Question

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.

Answer 1

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.