Question

Please help me, I'm really confused! I'll attach a picture below!!

A box is to be formed by cutting square pieces out of the corner of a rectangular piece of a 4" by 6" note card. The sides will then be folded up to make the box.

-how many pieces of length x have been cut from the original length?
-write the resulting length of the note card, once the pieces have been cut.
-how many pieces of length x have been cut from the original width?
-write the resulting width of the note card once the pieces have been cut.

Answer 1

`-how many pieces of length x have been cut from the original length?`
So if we assume that the 4'' is our length:
We're making 2 cuts along each side of the length.
So we'll have 4 total pieces of length x, 2 on each side of the length.
Unless they just want you to analyze one side, I'm not really sure...

`-write the resulting length of the note card, once the pieces have been cut.`
So our length was 4, and now we'll subtract a length x and another length x.
So our new length will be 4-2x.

Does that part make sense?

Answer 2

Yes! That makes sense!! @zepdrix

Have 4 original pieces of length x also been cut from the original width?
What would the resulting width be after how ever many pieces of length x have been cut from the original width?

Answer 3

Yah it looks like we're making the same adjustments to the width ( again I dunno if they want us to say 2 or 4 ). I would guess they're asking for how many cuts have been applied to each side of the width, which is 2. Blahhh I dunno :o

So if our width is 6 and we subtract x and x from that width, what expression can we set up for our new width?

Answer 4

6-2x? @zepdrix

Answer 5

Ya sounds good!


So we started with a `4'' by 6''` notecard.
By making cuts along the length and width we've determined the new dimensions (which depend on x) will be `(4-2x)'' by (6-2x)''`.

Answer 6

Thank you! Can I ask you just a few more questions in regards to the note card?

(If yes, then...)

-the area of the bottom of the box is the product of the new length and width. Write this area as a function in the form a(x)= your answer.

-Assume you want the bottom of your box to cover a total area of 16in^2. Set your question from part 2-(4-2x)" by (6-2x)"--equal to 16 and use the quadratic formula to solve for the two possible answers.

- one of these answers is not possible due to the original dimensions of the note card. Clearly indicate which answer is not possible. @zepdrix

Answer 7

We could find the area of the original `4''` by `6''` notecard by multiplying the length times the width. A = `4''` * `6''`

For our new notecard, `(4-2x)''` by `(6-2x)''` how will calculate the new area?
a(x) = `?` * `?`

Answer 8

a(x)= (4-2x)" x (6-2x)"?

Answer 9

Ya looks good!
We can probably remove the units when we're doing the multiplication, it'll look nicer that way.

a(x) = (4-2x) * (6-2x)

I should have done that in the example I gave :P heh

Answer 10

For the next part ~ Assume you want the bottom of your box to cover a total area of 16in^2.

So our area (which now depends on x), we'll set equal to 16.

16 = (4-2x) * (6-2x)

And then solve for x.
Do you remember how to multiply out the brackets?
Does FOIL or something like that sound familiar?

Answer 11

Yes! Foiling it out I get 4x^2+2x+24, I believe? @zepdrix

Answer 12

Hmm I think your middle terms got a little screwed up there.
-2x*4 = -8x
-2x*6 = -12x

Answer 13

Which gives us something like -20x for the middle term, yes?

Answer 14

Yes, I see my mistake now! So the foil would be: 4x^2-20x+24? @zepdrix

Answer 15

Yes.

Answer 16

So they want you to throw that into the `quadratic formula` to solve for x.
Remember how to do that?

Answer 17

Yes! And using the quadratic formula, I got the answers 70i and 1/2(5-7i), would that be correct? @zepdrix

Answer 18

Hmm we shouldn't end up with imaginary solutions... Let's see what went wrong.

x = [ -b ± sqrt( b^2 - 4ac ) ] / 2a

x = [ 20 ± sqrt( 400 - 384 ) ] / 8

4ac = 4(4)(24) = 384, yes?

Answer 19

Oh, I see! Sorry, I must have miscalculated! So would 384 and -384 be our answers? And would the negative one be the one that's not possible due to the dimensions of the note card? @zepdrix

Answer 20

384? D: No.. that's not the answer... that's under the root.
There are some steps to do after that.

400 - 384 = 16
So now we have a 16 under the root.

Answer 21

x = [ 20 ± sqrt( 16 ) ] / 8

16 is a perfect square, yes?
Taking it's square root gives us 4.

x = [ 20 ± 4 ] / 8

Answer 22

So evaluating that, the new answer would be 3 or-3? D;

Answer 23

No... :c

Answer 24

For the plus case,

x = [ 20 + 4 ] / 8
x = 24 / 8
x = 3

Ok good you got that one.

How bout for the negative case?

x = [ 20 - 4 ] / 8
x = 16 / 8
x = ?

Answer 25

x=2?

Answer 26

Ok good so we end up with x=2 and x=3.
We didn't end up with a negative, so it's not as obvious which one is the bad value.

Answer 27

If we take x=3 and plug it into our dimensions,

4-2x and 6-2x

do we run into any problems?

We can check x=2 afterwards if x=3 is fine.

Answer 28

Ohh no we made a boo boo way back! :(

Answer 29

Nooooooooooo :CCC

Answer 30

Oh no! Where?!

Answer 31

You expanded the brackets correctly:
4x^2-20x+24

But remember the other side was not 0, it was 16.
They told us to set it equal to 16.

So our expression is:
16 = 4x^2-20x+24

We need to subtract 16 from each side before we can apply the quadratic formula.

Answer 32

0 = 4x^2 - 20x + 8

So that's the expression we want to use in our quadratic formula.

Answer 33

x = [ -b ± sqrt( b^2 - 4ac ) ] / 2a

x = [ 20 ± sqrt( 400 - 4(4)(8) ) ] / 2(4)

x = [ 20 ± sqrt(272) ] / 8

Answer 34

x = [ 20 ± 16.5 ] / 8

Answer 35

Does this make any sense? D':

We can only apply the quadratic formula when the `other side` is zero.
I forgot to put the 16 on the left, which made me think we could do it immediately.

Answer 36

Yes, I think I'm catching on so far! @zepdrix Simplifying your last equation further, I find myself coming up with 22.06 and 17.94, any chance these are correct?

Answer 37

No not even close :(
Hmm you need to work on your algebra steps I think...

Look at each case separately,

x = [ 20 + 16.5 ] / 8
x = ?


x = [ 20 - 16.5 ] / 8
x= ?

Answer 38

Do the math inside of the square brackets first.

Answer 39

This time I got 3.5 and .4375? @zepdrix

Answer 40

Ok good!

Answer 41

Again, we didn't get any negative answer, so it's hard to tell which one is trouble.

Plug x=3.5 into your length

6-2x

does it cause any problems?

Answer 42

It comes out as -1, which isn't plausible? @zepdrix

Answer 43

Correct!
We can't have a box that has `negative` length, that doesn't make sense.

Answer 44

Got it! Thank you so much for all of your help!!! @zepdrix

Answer 45

no prob \c:/