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QuestionA mechanic earns \14/hourregularand$21/hourovertime.Find14/hour regular and \$21/hour overtime. Find W(30),, W(40),, W(45),, W(50)$ and new function for 36 hours.

Studdy Solution

STEP 1

Assumptions1. The mechanic's regular hourly pay is 14.00.Theovertimepayistime−and−a−half,whichmeansit′s1.5timestheregularpay3.Theregularworkweekis40hours4.Theweeklywagefunctionisgivenby14.00. The overtime pay is time-and-a-half, which means it's1.5 times the regular pay3. The regular work week is40 hours4. The weekly wage function is given by W(h)$5. The function is split into two parts one for hours less than or equal to40, and one for hours greater than40

STEP 2

We are asked to evaluate W(30),W(40),W(45)W(30), W(40), W(45), and W(50)W(50). Let's start with W(30)W(30).
Since30 is less than or equal to40, we use the first part of the functionW(h)=14hW(h) =14h

STEP 3

Plug in the value for hh to calculate W(30)W(30).
W(30)=14×30W(30) =14 \times30

STEP 4

Calculate the value of W(30)W(30).
W(30)=14×30=$420W(30) =14 \times30 = \$420

STEP 5

Next, we evaluate W(40)W(40). Since40 is equal to40, we again use the first part of the functionW(h)=14hW(h) =14h

STEP 6

Plug in the value for hh to calculate W(40)W(40).
W(40)=14×40W(40) =14 \times40

STEP 7

Calculate the value of W(40)W(40).
W(40)=14×40=$560W(40) =14 \times40 = \$560

STEP 8

Next, we evaluate W(45)W(45). Since45 is greater than40, we use the second part of the functionW(h)=21(h−40)+560W(h) =21(h-40) +560

STEP 9

Plug in the value for hh to calculate W(45)W(45).
W(45)=21×(45−40)+560W(45) =21 \times (45-40) +560

STEP 10

Calculate the value of W(45)W(45).
W(45)=21×(45−40)+560=$665W(45) =21 \times (45-40) +560 = \$665

STEP 11

Finally, we evaluate W(50)W(50). Since50 is greater than40, we use the second part of the functionW(h)=21(h−40)+560W(h) =21(h-40) +560

STEP 12

Plug in the value for hh to calculate W(50)W(50).
W(50)=21×(50−40)+560W(50) =21 \times (50-40) +560

STEP 13

Calculate the value of W(50)W(50).
W(50)=21×(50−40)+560=$770W(50) =21 \times (50-40) +560 = \$770

STEP 14

For part (b), we are asked to find the new weekly wage function if the regular work week is decreased to36 hours.
The new function will be similar to the old one, but the split between regular time and overtime will now happen at36 hours instead of40.

STEP 15

Write down the new weekly wage function. For hours less than or equal to36, the wage is still 14perhour.Forhoursgreaterthan36,thewageistime−and−a−half,or14 per hour. For hours greater than36, the wage is time-and-a-half, or 21 per hour, for the extra hours, plus the wage for the first36 hours.
Wnew(h)={14h,0<h≤3621(h−36)+504,h>36W_{new}(h)=\left\{\begin{array}{ll} 14h, &0<h \leq36 \\ 21(h-36)+504, & h>36\end{array}\right.

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