To determine the correct expression for the volume of a box formed by cutting out square corners of side length [tex]\( x \)[/tex] from a piece of cardboard, let's analyze each expression carefully.
When making a box from a flat piece of cardboard by cutting out square corners with side length [tex]\( x \)[/tex], the resulting dimensions of the box will be affected based on the size of the initial cardboard piece.
Let's consider that the starting size of the cardboard is 14 units by 22 units, as an example.
### Volume Formulas Analysis:
1. First Expression: [tex]\((x-7)(x-11)x\)[/tex]
- This implies the box will have length [tex]\((x-7)\)[/tex], width [tex]\((x-11)\)[/tex], and height [tex]\(x\)[/tex].
- This doesn't quite seem to fit our situation as both [tex]\(-7\)[/tex] and [tex]\(-11\)[/tex] would imply negative arguments.
2. Second Expression: [tex]\((7-2x)(11-2x)x\)[/tex]
- Here, the length is [tex]\((7-2x)\)[/tex], the width is [tex]\((11-2x)\)[/tex], and the height is [tex]\(x\)[/tex].
- When squares of side length [tex]\(x\)[/tex] are cut out of each corner, we subtract twice [tex]\(x\)[/tex] from both dimensions of the cardboard sheet.
3. Third Expression: [tex]\((7x-11)(7-11x)\)[/tex]
- This implies the box has dimensions [tex]\(7x-11\)[/tex] for one side and [tex]\(7-11x\)[/tex] for the other.
- These too could be negative or incorrectly scaled dimensions.
4. Fourth Expression: [tex]\((11-7x)(11x-7)\)[/tex]
- Similarly, this gives dimensions [tex]\(11-7x\)[/tex] and [tex]\(11x-7\)[/tex].
- Just like the third expression, these dimensions are troublesome as they don't match expectations for a basic cut-box setup.
### Validating the Correct Expression:
The formula for the volume of the box formed in this way, based on standard cutout procedures, would typically be:
- Length and width each have twice the cutout length removed from them, so the dimensions would be [tex]\((a - 2x)\)[/tex] and [tex]\((b - 2x)\)[/tex] respectively, and the height is [tex]\(x\)[/tex].
For the provided problem:
- Cut from length: [tex]\((initial\ length - 2x) = (7 - 2x)\)[/tex]
- Cut from width: [tex]\((initial\ width - 2x) = (11 - 2x)\)[/tex]
- The height of the box is [tex]\(x\)[/tex].
Thus, the second expression [tex]\((7-2x)(11-2x)x\)[/tex] fits the scenario of creating a box by cutting out squares from a rectangular piece of cardboard.
So, the correct expression to determine the greatest possible volume of the cardboard box is:
[tex]\[
(7-2x)(11-2x)x.
\][/tex]
