Final answer:
Lydia will earn an additional $92 by working 45 hours instead of 36 hours. To earn $500, Lydia needs to work a total of 55 hours, which includes overtime pay for the hours worked beyond 40.
Step-by-step explanation:
Lydia's weekly pay is a piecewise function, represented by S(h), where S(h) = 8h for hours less than or equal to 40 and S(h) = 12h - 160 for hours greater than 40.
a. If Lydia worked 36 hours and then adds a 9-hour shift, she will have worked 45 hours in total. For the first 40 hours, she earns 8 dollars per hour, and for the additional 5 hours (because 45 > 40), she earns at the overtime rate. Her extra earnings for working 45 hours instead of 36 can be calculated as follows:
- Earnings for 36 hours: 36 hours * $8/hour = $288
- Earnings for 40 hours: 40 hours * $8/hour = $320
- Earnings for additional 5 hours: 5 hours * $12/hour = $60
- Total earnings for 45 hours: $320 + $60 = $380
- Additional earnings: $380 - $288 = $92
b. To earn $500 at the given rates, Lydia can work up to 40 hours at $8/hour and then needs to work additional hours at an overtime rate of $12/hour. If her total earnings for 40 hours are $320, she needs an additional $500 - $320 = $180. Dividing this amount by the overtime rate:
Hours at overtime = $180 / $12/hour = 15 hours
Total hours to earn $500: 40 hours + 15 hours = 55 hours.