Part I
We have the size of the sheet of cardboard and we'll use the variable "x" to represent the length of the cuts. For any given cut, the available distance is reduced by twice the length of the cut. So we can create the following equations for length, width, and height.
width: w = 12 - 2x
length: l = 18 - 2x
height: h = x

Part II
v = l * w * h
v = (18 - 2x)(12 - 2x)x
v = (216 - 36x - 24x + 4x^2)x
v = (216 - 60x + 4x^2)x
v = 216x - 60x^2 + 4x^3
v = 4x^3 - 60x^2 + 216x

Part III
The length of the cut has to be greater than 0 and less than half the length of the smallest dimension of the cardboard (after all, there has to be something left over after cutting out the corners). So 0 < x < 6

Let's try to figure out an x that gives a volume of 224 in^3. Since this is high school math, it's unlikely that you've been taught how to handle cubic equations, so let's instead look at integer values of x. If we use a value of 1, we get a volume of:
v = 4x^3 - 60x^2 + 216x
v = 4*1^3 - 60*1^2 + 216*1
v = 4*1 - 60*1 + 216
v = 4 - 60 + 216
v = 160

Too small, so let's try 2.
v = 4x^3 - 60x^2 + 216x
v = 4*2^3 - 60*2^2 + 216*2
v = 4*8 - 60*4 + 216*2
v = 32 - 240 + 432
v = 224

And that's the desired volume.
So let's choose a value of x=2.
Reason?
It meets the inequality of 0 < x < 6 and it also gives the desired volume of 224 cubic inches.

3 0

3 years ago

Please help! :))

denis23 [38]

Answer:

Check below

Step-by-step explanation:

Hi,

We're dealing with linear functions. $f(x)=mx+b$

We have here the first function.

$g(x)=3x$

That's a linear function, with slope, and no linear coefficient since

But on the other hand, the functions below, they all describe parallel lines, since their slope has the same value. I've changed the letters to make it easier the comprehension.

Even though each one has the same slope value, they have different non zero linear coefficients, {-6,-18,6,18}. Unlike, the first one

$h(x)=3(9x-2) \rightarrow h(x)= 27x-6\\i(x)=9(3x-2)\rightarrow i(x)=27x-18\\j(x)=3(9x+2)\rightarrow j(x)=27x+6 \\k(x)=9(3x+2)\rightarrow k(x)=27x+18\\$