The problem here involves creating an open-top box by cutting out squares from each corner of a square sheet of cardboard, then folding up the sides. Here's how we can solve this step-by-step:

Step 1: Define Variables

1. Let the side length of the original square sheet of cardboard be $s$.
2. The small squares cut out from each corner are 2 inches on each side, so the depth (height) of the box will be $h = 2$ inches.

Step 2: Set up the Dimensions of the Box

After cutting out the corners, the new length and width of the box’s base will each be reduced by twice the side length of the cut squares (2 inches from each side), so:

$\text{Length and Width of the base} = s - 2 \cdot 2 = s - 4$

Thus, the base of the box has dimensions $(s - 4) \times (s - 4)$, and the height of the box is 2 inches.

Step 3: Volume of the Box

The volume $V$ of a rectangular box is given by:

$V = \text{Length} \times \text{Width} \times \text{Height}$

Substitute the values we know:

$128 = (s - 4) \times (s - 4) \times 2$

Step 4: Solve for $s$

1. Divide both sides by 2:

   $64 = (s - 4)^2$

2. Take the square root of both sides:

   $s - 4 = \pm 8$

3. Solve for $s$:

   • $s - 4 = 8 \Rightarrow s = 12$
   • $s - 4 = -8$ is not feasible as $s$ must be positive.

Thus, $s = 12$ inches.

Conclusion

The side length of the original sheet of cardboard is 12 inches.

Would you like a more detailed breakdown of any step?

Here are 5 related questions to explore this concept further:

1. How would the dimensions change if the cut squares had a different size?
2. What if the required volume of the box was different?
3. How would this problem change if the cardboard was a rectangle instead of a square?
4. What if only two squares were cut from opposite corners—would this still make a box?
5. How would adding a lid to the box change the calculations?

Tip: When dealing with geometric problems, start by defining all given dimensions and relationships clearly. This simplifies setting up equations to find unknowns.