To find out how much Garrett will make for the week, we can start by defining the piecewise function based on his pay structure:
Let \( P(x) \) be the amount Garrett earns for \( x \) hours worked in one day:
\[ P(x) = \begin{cases} 14x & \text{if } x \leq 8 \ 14 \cdot 8 + 16(x - 8) & \text{if } x > 8 \end{cases} \]
This function states:
- If Garrett works 8 hours or less (\( x \leq 8 \)), he earns a flat rate of $14 per hour.
- If Garrett works more than 8 hours (\( x > 8 \)), he earns $14 for the first 8 hours, plus $16 for each additional hour worked past 8 hours.
Example Table of Hours Worked
Let's say the table of hours Garrett worked each day for the week is as follows:
| Day | Hours Worked (x) | |-----------|------------------| | Monday | 7 | | Tuesday | 9 | | Wednesday | 8 | | Thursday | 10 | | Friday | 6 | | Saturday | 12 | | Sunday | 5 |
Calculate Weekly Earnings
Now, we'll use the piecewise function \( P(x) \) to calculate the amount Garrett earns each day based on the hours worked:
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Monday (7 hours): \[ P(7) = 14 \cdot 7 = 98 \]
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Tuesday (9 hours): \[ P(9) = 14 \cdot 8 + 16(9 - 8) = 112 + 16 = 128 \]
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Wednesday (8 hours): \[ P(8) = 14 \cdot 8 = 112 \]
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Thursday (10 hours): \[ P(10) = 14 \cdot 8 + 16(10 - 8) = 112 + 32 = 144 \]
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Friday (6 hours): \[ P(6) = 14 \cdot 6 = 84 \]
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Saturday (12 hours): \[ P(12) = 14 \cdot 8 + 16(12 - 8) = 112 + 64 = 176 \]
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Sunday (5 hours): \[ P(5) = 14 \cdot 5 = 70 \]
Total Earnings for the Week
Now, we can sum up the daily earnings:
\[ \text{Total Earnings} = P(7) + P(9) + P(8) + P(10) + P(6) + P(12) + P(5) \]
Substituting the earnings from each calculation:
\[ \text{Total Earnings} = 98 + 128 + 112 + 144 + 84 + 176 + 70 \]
Calculating the total:
\[ \text{Total Earnings} = 98 + 128 + 112 + 144 + 84 + 176 + 70 = 812 \]
Conclusion
Garrett will make a total of $812 for the week.
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