Question

a welder earns $20 an hour for the first 40 hours she works in a week and $30 for every hour over 40 hours. Determine a piece wise function f(x) that can be used to determine the welder's earnings when she works for x hours in a week

Answer 1

@DangerousJesse @aum @zepdrix @kohai

Answer 2

A female welder? 0_o

Answer 3

XD
Anyway let's seeeee... ummm..

Answer 4

Whooaa 0-o

Answer 5

If \(\Large\rm x\) is the number of hours she worked,
Then she earns \(\Large\rm 20x\) during the first 40 hours.

I guess another way we could think of this is,

\(\Large\rm 20x,\qquad x\le40\)
right?
She's making 20 dollars per x, as long as x is less than or equal to 40 hours.

Answer 6

That would be our first "piece".

Answer 7

ok..

Answer 8

Hmm the second piece is a little harder.
Let's, for a sec, assume she worked `exactly` 40 hours. How much money did she make?

Answer 9

\[30x, x >40\] is this the second one?

Answer 10

$800

Answer 11

This is what we have to be careful about with the second piece:
30x looks like she's earning 30 dollars an hour for ALL of the hours she's worked.

But she's only supposed to earn 30 dollars for the hours ABOVE 40.

Answer 12

So when she works more than 40 hours, she earns that 800 dollars (from the $20 rate), and then 30 dollars per hour for the additional hours on top of that.

Answer 13

They made this problem as difficult as possible lol.
It's gonna be a tricky one :)

Answer 14

springboard.... -_-

Answer 15

So your next thought might be "oh well then it's clearly 800+30x, right?"

Answer 16

But...

Answer 17

We still have the problem that we're paying her 30 dollars for ALL OF THE HOURS.

Answer 18

How can we fix that? c:

Answer 19

do we have to make it into its own parts?

Answer 20

Nahh, let's not try anything fancy like that.

Let's think of some numbers really quick.
If she works 41 hours, she earns the 800 from her previous rate `plus` 30 dollars for the next hour.

Answer 21

If she works 42 hours, she earns 800 plus 60 dollars, 30 for each of the 2 hours above 40.

Answer 22

So when she works 42 hours, she gets paid the 30 dollar rate for this many hours: \(\Large\rm 42-40\)

Or we could generalize it and say that she's getting paid the 30 dollar rate when she works
\(\Large\rm x-40\) hours.

Answer 23

What'dya think? too confusing? :c

Answer 24

ummm.. how would we put that into 1 equation?

Answer 25

x-40 doesnt make much sense to me

Answer 26

Well hold on, we're still trying to figure out the second piece.
Then we'll just smoosh the pieces together once we have them.

Answer 27

alright c:

Answer 28

This is what we're thinking our second piece should look like:\[\Large\rm 800+30(\color{orangered}{x})\]
But the orange x is the problem.
It says ~ she's getting paid $30/hr for EVERY hour that she works~.
So to fix that, we only want her to be paid $30/hr for every hour ABOVE 40.
So we're subtracting 40 from our x value.\[\Large\rm 800+30(\color{royalblue}{x-40})\]Now she's only getting paid $30/hr for the hours above 40.
Example: if we plug in x=3,
43-40 = 3
So she gets paid $30/hr for 3 of those 43 hours.

Answer 29

And then we specify that we're only using this piece when x is greater than 40 hours,
\[\Large\rm 800+30(x-40),\qquad x\gt40\]

Answer 30

that makes sense! :)

Answer 31

So to create our piece-wise function, we just group the pieces together with the fancy curly bracket!

Answer 32

\[\Large\rm f(x)=\cases{20x, &x≤40\\ \rm 800+30(x-40), &x>40}\]Something like that, yah? :o

Answer 33

Aw did it come out as a bunch of "question mark boxes" on your screen also? D:
Man I hate when that happens..

Answer 34

ya, but i know it meant less than or equal to

Answer 35

k cool c:

Phew, finally done.
Hopefully it made a bit of sense.
I told ya this one was gonna be tough! D:

Answer 36

it did! thanks for helping :D