Assume you work at a company where you are paid hourly. You are paid $8.60 per hour for regular time (less than or equal to forty hours) and time and a half for overtime hours up to fifty hours in one week. If you are asked to work fifty or more hours in one week you are paid double-overtime. Write this information in the form of a piecewise-defined function and be sure to include the domain for each piece. Also, check your model by calculating the pay for working forty hours and for working forty-one hours. Post these values as well.
To express this situation as a piecewise-defined function, we'll break it down into different cases based on the number of hours worked in one week.
Let's denote the number of hours worked in one week as h.
For h ≤ 40 (regular time), the pay rate is $8.60 per hour. So the pay can be represented by the function:
P(h) = 8.60h, where 0 ≤ h ≤ 40.
For 40 < h ≤ 50 (overtime), the pay rate is time and a half ($8.60 × 1.5 = $12.90) per hour. So the pay in this range can be represented by the function:
P(h) = 8.60(40) + 12.90(h - 40), where 40 < h ≤ 50.
For h > 50 (double-overtime), the pay rate is double ($8.60 × 2 = $17.20) per hour. So the pay in this range can be represented by the function:
P(h) = 8.60(40) + 12.90(50 - 40) + 17.20(h - 50), where h > 50.
Let's check the model by calculating the pay for working 40 hours and for working 41 hours:
For 40 hours worked, using the first piece of the function: P(40) = 8.60(40) = $344.
For 41 hours worked, using the second piece of the function: P(41) = 8.60(40) + 12.90(41 - 40) = $356.90.
So, for 40 hours worked, the pay is $344, and for 41 hours worked, the pay is $356.90.
The piecewise-defined function representing the pay structure at the company would be as follows:
f(x) = 8.60x, for 0 ≤ x ≤ 40
f(x) = (8.60 * 40) + (8.60 * 1.5 * (x - 40)), for 40 < x ≤ 50
f(x) = (8.60 * 40) + (8.60 * 1.5 * 10) + (8.60 * 2 * (x - 50)), for x > 50
In this function:
- The first piece, 8.60x, represents the regular pay rate for hours up to 40.
- The second piece, (8.60 * 40) + (8.60 * 1.5 * (x - 40)), represents the pay rate for overtime hours between 41 and 50.
- The third piece, (8.60 * 40) + (8.60 * 1.5 * 10) + (8.60 * 2 * (x - 50)), represents the pay rate for double-overtime hours above 50.
To calculate the pay for working 40 hours:
f(40) = 8.60 * 40 = $344
To calculate the pay for working 41 hours:
f(41) = (8.60 * 40) + (8.60 * 1.5 * (41 - 40))
= (344) + (8.60 * 1.5 * 1)
= 344 + 12.90
= $356.90
So, the pay for working 40 hours is $344, and the pay for working 41 hours is $356.90.