Math Problem Statement
Nate's weekly pay, in dollars, when he works x hours is represented by a piecewise function. How much is Nate paid for each hour he works over 40?
Solution
To solve this problem, we need to find how much Nate is paid for each hour he works over 40 hours.
Step-by-Step Solution
The piecewise function given is:
\begin{cases} 16.50x & \text{if } 0 \leq x \leq 40 \\ 24.75(x - 40) + 660 & \text{if } x > 40 \end{cases}$$ - **First part**: $$16.50x$$ is the pay when Nate works between 0 to 40 hours. - **Second part**: $$24.75(x - 40) + 660$$ is the pay when Nate works more than 40 hours. The term $$24.75(x - 40)$$ represents the additional amount Nate earns for hours over 40. 1. **Interpretation of the expression:** - $$x - 40$$ is the number of hours worked over 40. - $$24.75$$ is the rate Nate is paid for each hour over 40. ### Conclusion Nate is paid **$24.75** for each hour he works over 40. ### Correct Answer The correct answer is **B. $24.75**. Would you like further details or have any questions about this solution? --- Here are 5 related questions to explore further: 1. How would Nate's pay change if the rate for hours over 40 increased to $30 per hour? 2. What would be Nate's total weekly pay if he worked 45 hours in a week? 3. How does the piecewise function ensure a smooth transition in payment at 40 hours? 4. Can we find a general formula for calculating Nate's pay if the rates change? 5. What are other real-world examples where piecewise functions might be useful? **Tip:** When solving piecewise functions, always carefully check the conditions under which each part of the function applies.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Rates of Change
Formulas
Pay for hours over 40 is calculated using 24.75(x - 40) + 660 where x > 40.
Theorems
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Suitable Grade Level
Grades 9-10