Answer :
To determine which piecewise equation correctly models the technician's weekly pay [tex]\( y \)[/tex] as it relates to the number of hours [tex]\( x \)[/tex] worked, we need to understand the pay structure:
1. For the first 40 hours, the technician is paid [tex]$25 per hour. 2. For any hours worked beyond 40, the technician is paid $[/tex]32 per hour for those additional hours.
Let's analyze each given option:
Option A:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & \text{if } 0 \leq x \leq 40 \\ 32(x-40) & \text{if } x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is calculated as [tex]\( 25x \)[/tex], where [tex]\( x \)[/tex] is the number of hours worked. This is correct because the pay rate is [tex]$25 per hour. For \( x > 40 \): - The pay for the first 40 hours is \( 25 \times 40 \). - The pay for the hours beyond 40 is \( 32 \times (x - 40) \), where \( (x - 40) \) represents the hours worked beyond 40 hours. Combining these, the total pay becomes: \[ y = 25 \times 40 + 32 \times (x - 40) \] \[ y = 1000 + 32(x - 40) \] Simplifying further: \[ y = 1000 + 32x - 1280 \] \[ y = 32x - 280 \] So, Option A indeed models the pay correctly for the conditions given. Option B: \[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32x + 1000 & x > 40 \end{array} \right.\] For \( 0 \leq x \leq 40 \): - The weekly pay \( y \) is \( 25x \), which is correct. For \( x > 40 \): - The form \( 32x + 1000 \) is problematic. This suggests a different base pay and doesn't properly account for the first 40 hours already compensated at $[/tex]25/hour. This equation does not correctly model the weekly pay structure given.
Option C:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32x & x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is [tex]\( 25x \)[/tex], which is correct.
For [tex]\( x > 40 \)[/tex]:
- This assumes all hours, including the first 40, are paid at $32 per hour which is incorrect. It does not separate the two different pay rates properly.
Option D:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32(x-40) + 1000 & x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is [tex]\( 25x \)[/tex], which is correct.
For [tex]\( x > 40 \)[/tex]:
- The equation [tex]\( 32(x-40) + 1000 \)[/tex] compensates for the first 40 hours at [tex]\( 25 \times 40 = 1000 \)[/tex] and correctly adds the pay for extra hours. Therefore, this also accurately models the pay structure.
However, upon closer inspection, Option A correctly simplifies into the appropriate model:
- For [tex]\( 0 \leq x \leq 40 \)[/tex]: [tex]\( y = 25x \)[/tex]
- For [tex]\( x > 40 \)[/tex]: [tex]\( y = 32(x-40) + 1000 \)[/tex].
Combining for [tex]\( x > 40 \)[/tex]:
[tex]\[ y = 1000 + 32(x - 40) \][/tex]
[tex]\[ y = 1000 + 32x - 1280 \][/tex]
[tex]\[ y = 32x - 280 \][/tex]
Thus, Option D also correctly represents the pay structure. However, as a choice is required, and typical simple representation disambiguation, as Option A structure embodies the form for piecewise clear seeing between simplified visual, Option D is exact full form pay stratification reality.
Therefore, the most simplified correct choice representation is:
[tex]\[ \text{Correct Answer: A} \][/tex]
1. For the first 40 hours, the technician is paid [tex]$25 per hour. 2. For any hours worked beyond 40, the technician is paid $[/tex]32 per hour for those additional hours.
Let's analyze each given option:
Option A:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & \text{if } 0 \leq x \leq 40 \\ 32(x-40) & \text{if } x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is calculated as [tex]\( 25x \)[/tex], where [tex]\( x \)[/tex] is the number of hours worked. This is correct because the pay rate is [tex]$25 per hour. For \( x > 40 \): - The pay for the first 40 hours is \( 25 \times 40 \). - The pay for the hours beyond 40 is \( 32 \times (x - 40) \), where \( (x - 40) \) represents the hours worked beyond 40 hours. Combining these, the total pay becomes: \[ y = 25 \times 40 + 32 \times (x - 40) \] \[ y = 1000 + 32(x - 40) \] Simplifying further: \[ y = 1000 + 32x - 1280 \] \[ y = 32x - 280 \] So, Option A indeed models the pay correctly for the conditions given. Option B: \[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32x + 1000 & x > 40 \end{array} \right.\] For \( 0 \leq x \leq 40 \): - The weekly pay \( y \) is \( 25x \), which is correct. For \( x > 40 \): - The form \( 32x + 1000 \) is problematic. This suggests a different base pay and doesn't properly account for the first 40 hours already compensated at $[/tex]25/hour. This equation does not correctly model the weekly pay structure given.
Option C:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32x & x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is [tex]\( 25x \)[/tex], which is correct.
For [tex]\( x > 40 \)[/tex]:
- This assumes all hours, including the first 40, are paid at $32 per hour which is incorrect. It does not separate the two different pay rates properly.
Option D:
[tex]\[ y = \left\{ \begin{array}{ll} 25x & 0 \leq x \leq 40 \\ 32(x-40) + 1000 & x > 40 \end{array} \right.\][/tex]
For [tex]\( 0 \leq x \leq 40 \)[/tex]:
- The weekly pay [tex]\( y \)[/tex] is [tex]\( 25x \)[/tex], which is correct.
For [tex]\( x > 40 \)[/tex]:
- The equation [tex]\( 32(x-40) + 1000 \)[/tex] compensates for the first 40 hours at [tex]\( 25 \times 40 = 1000 \)[/tex] and correctly adds the pay for extra hours. Therefore, this also accurately models the pay structure.
However, upon closer inspection, Option A correctly simplifies into the appropriate model:
- For [tex]\( 0 \leq x \leq 40 \)[/tex]: [tex]\( y = 25x \)[/tex]
- For [tex]\( x > 40 \)[/tex]: [tex]\( y = 32(x-40) + 1000 \)[/tex].
Combining for [tex]\( x > 40 \)[/tex]:
[tex]\[ y = 1000 + 32(x - 40) \][/tex]
[tex]\[ y = 1000 + 32x - 1280 \][/tex]
[tex]\[ y = 32x - 280 \][/tex]
Thus, Option D also correctly represents the pay structure. However, as a choice is required, and typical simple representation disambiguation, as Option A structure embodies the form for piecewise clear seeing between simplified visual, Option D is exact full form pay stratification reality.
Therefore, the most simplified correct choice representation is:
[tex]\[ \text{Correct Answer: A} \][/tex]
