Find answers to your questions and expand your knowledge with IDNLearn.com. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.
Sagot :
To solve the problem and determine the appropriate piecewise equation for the engineering technician's weekly pay, let's break down the situation step by step:
1. Understanding the Pay Structure:
- For the first 40 hours worked in a week, the technician earns [tex]$\$[/tex]25[tex]$ per hour. - For any hours worked over 40 in a week, the technician earns $[/tex]\[tex]$32$[/tex] per hour for those extra hours.
2. Determining the Equations for Different Scenarios:
- Scenario 1: If the technician works between 0 and 40 hours (inclusive), the weekly pay [tex]\( y \)[/tex] is simply the number of hours [tex]\( x \)[/tex] multiplied by the standard hourly rate of [tex]$\$[/tex]25[tex]$. - Mathematically, this can be expressed as: \[ y = 25x \quad \text{for} \quad 0 \leq x \leq 40 \] - Scenario 2: If the technician works more than 40 hours, the weekly pay \( y \) includes two components: - Payment for the first 40 hours at the standard rate of $[/tex]\[tex]$25$[/tex] per hour.
- Payment for any additional hours (over 40) at the overtime rate of [tex]$\$[/tex]32$ per hour.
Mathematically, this can be expressed as:
- Payment for the first 40 hours: [tex]\( 25 \times 40 \)[/tex]
- Payment for additional hours (overtime [tex]\( x - 40 \)[/tex]): [tex]\( 32 \times (x - 40) \)[/tex]
Combining these components, the equation for the pay when [tex]\( x > 40 \)[/tex] is:
[tex]\[ y = 32(x - 40) + 25 \times 40 \][/tex]
Simplifying [tex]\( 25 \times 40 \)[/tex] gives us 1000, so:
[tex]\[ y = 32(x - 40) + 1000 \quad \text{for} \quad x > 40 \][/tex]
3. Matching the Equations with the Given Options:
The piecewise function needs to cover both scenarios accurately.
Comparing our derived equations with the given options:
- Option A: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32(x - 40) \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because it does not include the base pay of 1000 for hours over 40.
- Option B: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32x \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because the second part doesn't correctly account for the first 40 hours of standard pay.
- Option C: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32x + 1000 \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because the term [tex]\(32x + 1000\)[/tex] overestimates the pay for hours beyond 40.
- Option D: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32(x - 40) + 1000 \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Correct as it matches exactly with our derived equations. It accurately represents the weekly pay calculation for both scenarios.
Hence, the correct piecewise equation is:
[tex]\[ \boxed{D} \][/tex]
1. Understanding the Pay Structure:
- For the first 40 hours worked in a week, the technician earns [tex]$\$[/tex]25[tex]$ per hour. - For any hours worked over 40 in a week, the technician earns $[/tex]\[tex]$32$[/tex] per hour for those extra hours.
2. Determining the Equations for Different Scenarios:
- Scenario 1: If the technician works between 0 and 40 hours (inclusive), the weekly pay [tex]\( y \)[/tex] is simply the number of hours [tex]\( x \)[/tex] multiplied by the standard hourly rate of [tex]$\$[/tex]25[tex]$. - Mathematically, this can be expressed as: \[ y = 25x \quad \text{for} \quad 0 \leq x \leq 40 \] - Scenario 2: If the technician works more than 40 hours, the weekly pay \( y \) includes two components: - Payment for the first 40 hours at the standard rate of $[/tex]\[tex]$25$[/tex] per hour.
- Payment for any additional hours (over 40) at the overtime rate of [tex]$\$[/tex]32$ per hour.
Mathematically, this can be expressed as:
- Payment for the first 40 hours: [tex]\( 25 \times 40 \)[/tex]
- Payment for additional hours (overtime [tex]\( x - 40 \)[/tex]): [tex]\( 32 \times (x - 40) \)[/tex]
Combining these components, the equation for the pay when [tex]\( x > 40 \)[/tex] is:
[tex]\[ y = 32(x - 40) + 25 \times 40 \][/tex]
Simplifying [tex]\( 25 \times 40 \)[/tex] gives us 1000, so:
[tex]\[ y = 32(x - 40) + 1000 \quad \text{for} \quad x > 40 \][/tex]
3. Matching the Equations with the Given Options:
The piecewise function needs to cover both scenarios accurately.
Comparing our derived equations with the given options:
- Option A: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32(x - 40) \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because it does not include the base pay of 1000 for hours over 40.
- Option B: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32x \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because the second part doesn't correctly account for the first 40 hours of standard pay.
- Option C: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32x + 1000 \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Incorrect because the term [tex]\(32x + 1000\)[/tex] overestimates the pay for hours beyond 40.
- Option D: [tex]\( y = \left\{ \begin{array}{l} 25x \\ 32(x - 40) + 1000 \end{array} \right. \quad 0 \leq x \leq 40 \\ x > 40 \)[/tex]
- Correct as it matches exactly with our derived equations. It accurately represents the weekly pay calculation for both scenarios.
Hence, the correct piecewise equation is:
[tex]\[ \boxed{D} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
