Let’s solve this step by step.
We’re told:
- Regular pay rate = $7.00 per hour.
- Overtime (after 40 hours) is paid at “time and a half” → that means 1.5 times the regular rate.
- So overtime rate = 1.5 × $7.00 = $10.50 per hour.
Part a: Write a piecewise function.
Let x = number of hours worked.
Let f(x) = total money earned.
If you work 40 hours or less, you get $7/hour for all hours.
→ So if x ≤ 40, then f(x) = 7x.
If you work more than 40 hours:
- First 40 hours are paid at $7/hour → 40 × 7 = $280.
- Any hours over 40 (that’s x - 40) are paid at $10.50/hour → 10.50 × (x - 40).
→ So if x > 40, then f(x) = 280 + 10.50(x - 40).
So the piecewise function is:
f(x) =
{
7x, if x ≤ 40
280 + 10.50(x - 40), if x > 40
}
Part b: How much do you make if you work 45 hours?
Since 45 > 40, we use the second part of the function.
f(45) = 280 + 10.50 × (45 - 40)
= 280 + 10.50 × 5
= 280 + 52.50
= 332.50
Double-check:
- 40 hours @ $7 = $280
- 5 hours overtime @ $10.50 = $52.50
Total = $280 + $52.50 = $332.50
✔
Final Answer:
a. f(x) = { 7x if x ≤ 40; 280 + 10.50(x - 40) if x > 40 }
b. $332.50
Parent Tip: Review the logic above to help your child master the concept of piecewise functions word problems worksheet.
