To determine the greatest possible volume of the cardboard box made from a rectangular piece of cardboard with dimensions 8 ft by 6 ft, we need to follow these steps:
1. Understand the problem setup:
- We start with a rectangular piece of cardboard that is 8 ft by 6 ft.
- We cut out squares with side length [tex]\( x \)[/tex] from each corner.
- After cutting out the squares, we fold up the resulting flaps to form the sides of the box.
2. Formulate the dimensions of the box:
- The new length of the base of the box after the squares are cut out is [tex]\( 8 - 2x \)[/tex] (as [tex]\( x \)[/tex] is cut from both sides).
- The new width of the base of the box becomes [tex]\( 6 - 2x \)[/tex].
- The height of the box is equal to the side length of the squares that were cut out, which is [tex]\( x \)[/tex].
3. Expression for the volume of the box:
- The volume [tex]\( V \)[/tex] of a rectangular prism (which our box is) is calculated by the product of length, width, and height. Thus, the volume [tex]\( V \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[
V = \text{length} \times \text{width} \times \text{height} = (8 - 2x)(6 - 2x)x
\][/tex]
4. Identifying the correct option:
- The given expressions are:
- [tex]\( (x-8)(x-6)x \)[/tex]
- [tex]\( (8x-2)(2x-6) \)[/tex]
- [tex]\( (8-2x)(6-2x)x \)[/tex]
- [tex]\( (x-8)(x-6)x \)[/tex]
- Comparing them with our derived volume expression [tex]\((8 - 2x)(6 - 2x)x\)[/tex], we see that the correct volume expression matches one of the options exactly.
Therefore, the correct expression that can be used to determine the greatest possible volume of the cardboard box is:
[tex]\[
(8-2x)(6-2x)x
\][/tex]
