Question

Piecewise function. A house painter charges $12 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week?

Answer 1

First 40 hours rate = $12 per hour.

Next 10 hours charge ( Upto 50 hours) = A half of $12 that is $6 per hour.

And after next 10 hours chrges ( More than 50 hours) = 2 times of initial rate, that is 2*12 = $24 per hours.

Let us take variale x for number of hours and f(x) is the function for total earning for x hours.

So, we can setup a piecewise function as

        $f(x)=\left \{ {{12x \ \ \ \ 0<= x <= 40}  \atop {6x \ \ \ \ 40<x<=50}}  \atop {24x \ \ \  \ \  50 < x}  \right.$

Therefore,

70 hours can be divided into = 40 hours + 10 hours + 20 hours.

Total earning of 70-hour week =  12* 40 hours  + 6 * 10 hours + 24*20 hours

= 480 + 60 + 480

= $1020.

Therefore, he earn $1020 for a 70-hour week.

Answer 2

Given that a house painter charges $ 12 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that, to determine how much does he earn for a 70-hour week The following calculation must be carried out, proposing a linear function:

• (40 x 12) + (10 x 12 x 1.5) + (20 x 12 x 2) = X
• 40 x 12 + 10 x 18 + 20 x 24 = X
• 480 + 180 + 480 = X
• 1,140 = X

Therefore, the house painter earns $ 1,140 for a 70-hour week.

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