Question

A local store owner pays her employees time-and-a-half for overtime. That means if an employee works more than 40 hours per week, the store will pay him 1.5 times his regular hourly wage of $6.50

Answer 1

(a) Write and graph a piecewise function that illustrates an employee's weekly pay, W, as a function of the number of hours, h, that employee works

Answer 2

remind me what a piecewise function is

Answer 3

jk, found it
http://www.mathsisfun.com/sets/functions-piecewise.html

Answer 4

Maybe this will help you... someone tried this question before http://openstudy.com/study#/updates/516b671ce4b02ec89c5aac8b

Answer 5

weekly pay, w will be our 'vertical axis'
number of hours, h will be our 'horizontal axis'

so the equation for the employee working 0-40 hours would be a straight line:
y=mx+b or
p=mh + b
since regular wage is 6.5, then that is our slope, m
we can figure out the 'b' term becuase it represens the y-intercept of the line. we know that if they work zero hours, they get paid zero dollars. so b=0
thus
p = 6.5h + b

that is the first part of the piece wise function

Answer 6

I thought a piecewise function has < or > in it

Answer 7

the second part will consist of the idea of being paid (1.5)6.5 dollars each hour over 40 hours.
so this part of the function will start at 40 hours, and have a slope of 1.5*6.5 which is 6.5+3.25 = 9.75 = m
so far the equation would be
p=9.75h + b
we now need to figure out what 'b' is.
we know that at 41hrs we will be paid 9.75*41 = 399.75,
from this knowledge we can figure out the point (41, 399.75),
plug this into our equation:
p=9.75h + b
and solve for 'b'

Answer 8

"I thought a piecewise function has < or > in it"
u are right, we will get to this part next

Answer 9

any questions so far on how to get the line equations for each function made so far?

Answer 10

no, not yet

Answer 11

so we have the two eqauations:
f(h) = 6.5h for 0<= h <= 40
f(h) = 9.75h + b for 40<h

*dont forget, 'b' needs to be solved for like i mentioned

so here are our piecewise functions using < and >

Answer 12

Ok I get that

Answer 13

so how do we get b?

Answer 14

take the equation:
p=9.75h + b
and plug in the point, (41, 399.75)
then solve the equation for 'b'

Answer 15

idk how to do that... p=9.75(41,399.75)+b? Also are you 100% on how to do this. I really need to get this right

Answer 16

p=9.75(41,399.75)+b
close, like this:
399.75 = 9.75(41) + b

Answer 17

Oh I thought you meant like 41,399 thousand

Answer 18

b=0?

Answer 19

correct ^_^

Answer 20

can you finish out the rest?
or are there questions you have about what we did?

Answer 21

Well just to clarify, what would I write as my answer for part a)?

Answer 22

well, for the function we would have:
f(h) = 6.5h for 0<= h <= 40
f(h) = 9.75h + b for 40<h

as for the paragraph, you'll want to say something about how we made one function for 0 to 40 hours becuase the slope, or wage, was the same value. and we made a second function for h>40 hours becuase from here the wage changes from 6.5 to 9.75 becuase of the 'time and a half' for overtime.

Answer 23

9.75h+b or 9.75h + 0? Also, are you 100% sure on these? Also again (sorry) I have to actually draw a graph

Answer 24

oh, just do 9.75h

Answer 25

yep, I am 100% sure, there is no other way possible we could do this

Answer 26

how do I plot a graph? If you could show me or tell me how that would be amazing! I'll be back in an hour

Answer 27

"we know that at 41hrs we will be paid 9.75*41 = 399.75" may not be accurate.
He will get paid the overtime rate only for the hours exceeding 40 hours a week.
So if he works 41 hours, he will get paid: (6.50 * 40) + (1 * 9.75) = 260 + 9.75 = $269.75

Answer 28

hey @ranga ! Are these piecewise functions correct? And how do I graph them?

Answer 29

W(h) = 6.5h for 0<= h <= 40
For hours exceeding 40 hours a week, the pay will be:
(6.5 * 40) + (h - 40) * 9.75 = 260 + 9.75h - 390 = 9.75h - 130
W(h) = 9.75h - 130 for h > 40

The piecewise function is:
W(h) = 6.5h for 0<= h <= 40
W(h) = 9.75h - 130 for h > 40

Answer 30

@ranga thanks for the clean up of my mess ^_^

Answer 31

Thanks to you @DemolisionWolf. You did most of the work.

Answer 32

@ranga haha,
I get sloppy the more time I sit on a problem!

Answer 33

So using the piecewise functions as a base (W(h) = 6.5h for 0<= h <= 40
W(h) = 9.75h - 130 for h > 40
Determine how much an employee will be paid if he were to work 50 hours in a week.

Answer 34

h = 50 hours is more than 40 and so we have to use the second equation:
W(h) = 9.75h - 130 for h > 40
Put h = 50 and calculate W(50)

Answer 35

357.5

Answer 36

Yes, $357.50

Answer 37

Oh ok, so that is my piecewise functions and graph? Thanks a lot both of you! I have my exam in 3 days so I'm trying to finish up the study guide. I have a few more questions today. Will you be available to help Ranga?

Answer 38

I will be here on and off as time permits. If you have any open questions when I am on I will try to answer.

Answer 39

Got it! Oh and why do you have -130 when demo wolf had +b?

Answer 40

In an earlier reply I show how I derived the equation.