To determine which piecewise equation models the total weekly pay [tex]\( y \)[/tex] for the number of hours [tex]\( x \)[/tex] that the engineering technician works, we need to consider two cases: when [tex]\( x \)[/tex] is between 0 and 40 hours, and when [tex]\( x \)[/tex] exceeds 40 hours.
1. Case 1: [tex]\( 0 \leq x \leq 40 \)[/tex]
- For any number of hours [tex]\( x \)[/tex] within this range, the technician earns [tex]$25 per hour.
- Therefore, the total weekly pay \( y \) can be calculated using the equation:
\[
y = 25x
\]
2. Case 2: \( x > 40 \)
- For the first 40 hours, the technician earns $[/tex]25 per hour, which totals:
[tex]\[
25 \times 40 = 1000 \text{ dollars}
\][/tex]
- For any additional hours beyond the initial 40 hours, she earns $32 per hour. If [tex]\( x \)[/tex] is the total number of hours worked in the week, the additional hours are [tex]\( x - 40 \)[/tex].
- Therefore, for these additional hours, the earnings would be:
[tex]\[
32 \times (x - 40) \text{ dollars}
\][/tex]
- Adding the earnings from the first 40 hours and the additional hours gives the total pay [tex]\( y \)[/tex]:
[tex]\[
y = 1000 + 32(x - 40)
\][/tex]
Combining these two cases into a single piecewise-defined function gives us:
[tex]\[
y = \begin{cases}
25x & \text{if } 0 \leq x \leq 40 \\
32(x - 40) + 1000 & \text{if } x > 40
\end{cases}
\][/tex]
Considering the given options:
- Option A: [tex]\( y=\left\{\begin{array}{ll} 25 x & 0 \leq x \leq 40 \\ 32(x-40)+1000 & x>40 \end{array}\right. \)[/tex]
Based on this step-by-step reasoning, the correct answer is Option A.
