A worker gets 18 dollars per hour for a normal work week of 40 hours and time-and-a-half for overtime. Assuming he works at least 40 hours a week, construct a function describing his weekly paycheck,P, as a function of the number of hours worked, t.
P(t)=____ dollards
I PUT P(t)= 720+30(t-40)
BUT IT SAID IT'S WRONG. HELP PLEASE :((
The first 40 hours are paid at $18/hour.
For any amount of time over that, the rate is 18 * 1.5 = $27
So:
P(t) = 18 * 40 + 27 * ( t - 40 ) =
720 + 27 t - 27 * 40 =
720 + 27 t - 1080 =
27 t - 360
To construct a function describing the worker's weekly paycheck, we need to consider the different pay rates for regular hours and overtime hours.
Given that the worker earns $18 per hour for regular hours and time-and-a-half for overtime, we can break down the calculation into two parts:
1. Calculate the pay for regular hours:
The worker earns $18 per hour for the first 40 hours. So, for any hours worked up to 40, the pay for those hours is obtained by multiplying the number of hours worked by the regular pay rate of $18 per hour.
2. Calculate the pay for overtime hours:
For any hours worked above 40, the worker earns time-and-a-half, which means the pay rate is $18 * 1.5 = $27 per overtime hour. So, for any hours worked above 40, the pay for those hours is obtained by multiplying the number of overtime hours by the overtime pay rate of $27 per hour.
Now, let's construct the function to calculate the weekly paycheck, P(t), as a function of the number of hours worked, t:
P(t) = (Regular hours pay) + (Overtime pay)
Regular hours pay = $18 * (number of regular hours)
Overtime pay = $27 * (number of overtime hours)
To determine the number of regular hours and overtime hours, we need to consider that the worker works at least 40 hours in a week. So, if the number of hours worked, t, is less than or equal to 40, then all hours are regular hours, and there are no overtime hours. Otherwise, if the number of hours worked, t, is more than 40, the difference (t - 40) represents the overtime hours.
Finally, we can write the function as:
P(t) = $18 * min(t, 40) + $27 * max(t - 40, 0)
This function considers both the regular pay rate for up to 40 hours and time-and-a-half pay rate for any overtime worked.
Simplifying further, if we expand the min and max functions:
P(t) = $18 * min(t, 40) + $27 * max(t - 40, 0)
= $18 * (t if t ≤ 40; 40 if t > 40) + $27 * (0 if t ≤ 40; (t - 40) if t > 40)
As a result, the correct function describing the worker's weekly paycheck is:
P(t) = $18 * min(t, 40) + $27 * max(t - 40, 0)
This function will give you the correct paycheck amount for different numbers of hours worked, t.