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 the regular rate for these overtime hours (those past the first 50 hours). 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.
p =
} 8.60h if h <= 40
} 344 + 12.9h if 40<h<=50
} 473 + 2h if h>50
Third equation should read:
473 + 17.2h if h>50
For forty-one hours, h=41, so 40<h≤50,
use the second equation.
The piecewise-defined function to calculate the pay per week can be written as follows:
Pay(hours) =
$8.60 * hours if 0 ≤ hours ≤ 40
$8.60 * 40 + ($8.60 * 1.5) * (hours - 40) if 40 < hours ≤ 50
$8.60 * 40 + ($8.60 * 1.5 * 10) + ($8.60 * 2) * (hours - 50) if hours > 50
The domain for each piece is as follows:
- For the first piece, 0 ≤ hours ≤ 40 as it represents the regular time pay rate.
- For the second piece, 40 < hours ≤ 50 as it calculates the pay for overtime hours up to 50 hours.
- For the third piece, hours > 50 as it calculates the pay for overtime hours beyond 50.
To check the model, let's calculate the pay for working 40 hours:
Pay(40) = $8.60 * 40 = $344.00
Now, let's calculate the pay for working 41 hours:
Pay(41) = $8.60 * 40 + ($8.60 * 1.5) * (41 - 40) = $344.00 + $12.90 = $356.90
Therefore, the pay for working 40 hours is $344.00, and the pay for working 41 hours is $356.90.
To write this information as a piecewise-defined function, we can consider the different pay rates based on the number of hours worked.
Let's denote the number of hours worked as "x".
For regular time (less than or equal to 40 hours), you are paid $8.60 per hour. So the pay for regular time can be represented as:
f(x) = 8.60x, where 0 <= x <= 40.
For overtime hours (41 to 50 hours), you are paid time and a half, which means 1.5 times the regular rate. The pay for overtime hours can be represented as:
f(x) = 1.5 * 8.60 * (x - 40), where 41 <= x <= 50.
For hours beyond 50, you are paid double the regular rate. The pay for these overtime hours can be represented as:
f(x) = 2 * 8.60 * (x - 50), where x > 50.
It's important to note that the pay model assumes the hourly rate remains consistent during regular time, overtime hours up to 50, and beyond 50 hours.
To check this model, let's calculate the pay for working 40 hours and 41 hours:
For 40 hours (regular time), we use the first piece of the function:
f(40) = 8.60 * 40 = $344.
For 41 hours (overtime), we use the second piece of the function:
f(41) = 1.5 * 8.60 * (41 - 40) = $12.9.
So, the pay for working 40 hours is $344, and the pay for working 41 hours is $12.9 (which includes some overtime pay).
Remember, this is a simplified model assuming that the pay rate and rules mentioned are fixed. In reality, different companies may have their own specific policies and additional considerations.