To find the correct piecewise equation that models the business analyst's total weekly pay [tex]\( y \)[/tex] as it relates to the number of hours [tex]\( x \)[/tex] worked, we will break down the problem step by step:
1. Determine the pay rate for the first 42 hours:
- The analyst earns [tex]$20 per hour for the first 42 hours.
- So, if \( 0 \leq x \leq 42 \), the total pay \( y \) is given by:
\[
y = 20x
\]
2. Determine the pay rate for hours worked beyond 42 hours:
- For hours worked beyond 42, the analyst earns $[/tex]28 per hour.
- This rate applies only to the hours exceeding 42.
- Therefore, if [tex]\( x \)[/tex] hours are worked where [tex]\( x > 42 \)[/tex]:
- The analyst earns $20 per hour for the first 42 hours, so the pay for these hours is [tex]\( 20 \times 42 \)[/tex].
- The total pay for the first 42 hours is:
[tex]\[
20 \times 42 = 840 \text{ dollars}
\][/tex]
- For the hours exceeding 42, let [tex]\( x - 42 \)[/tex] be the number of overtime hours.
- The pay for the overtime hours is [tex]\( 28 \times (x - 42) \)[/tex].
- Hence, the total pay [tex]\( y \)[/tex] for [tex]\( x > 42 \)[/tex] is:
[tex]\[
y = 28(x - 42) + 840
\][/tex]
3. Formulate the piecewise function:
- Combining the above pay calculations into the piecewise function, we get:
[tex]\[
y = \left\{\begin{array}{ll}
20x & \text{if } 0 \leq x \leq 42 \\
28(x-42) + 840 & \text{if } x > 42
\end{array}\right.
\][/tex]
The correct piecewise equation that models the analyst's total weekly pay is then:
[tex]\[
A. \quad y = \left\{\begin{array}{ll}
20x & 0 \leq x \leq 42 \\
28(x-42) + 840 & x > 42
\end{array}\right.
\][/tex]
