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The problem is asking us to find the original dimensions of a square sheet of cardboard used to create an open box by cutting out squares from each corner. The key information provided includes:

- The volume of the box is 578 cubic inches.
- The squares cut from each corner have a side length of 2 inches (as indicated in the diagram).
- The resulting open box will have height \(2 \, \text{inches}\) (from the square cuts) and the remaining dimensions (length and width) will depend on the original size of the square sheet.

Let the original side length of the square sheet of cardboard be \(x\). After cutting out squares of \(2 \, \text{inches}\) from each corner, the length and width of the base of the box will be \(x - 4\) (since two 2-inch sections are removed from each dimension).

The formula for the volume of the box is:
\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]
Substitute the known values:
\[
578 = (x - 4)(x - 4)(2)
\]
Simplifying:
\[
578 = 2(x - 4)^2
\]
Now, divide both sides by 2:
\[
289 = (x - 4)^2
\]
Take the square root of both sides:
\[
x - 4 = \sqrt{289}
\]
\[
x - 4 = 17
\]
Therefore:
\[
x = 17 + 4 = 21
\]

The original side length of the square sheet of cardboard is \(x = 21 \, \text{inches}\).

Would you like a detailed breakdown of the individual steps or further clarification on any part of the solution?

Here are 5 additional related questions to consider:
1. How would the equation change if the cut squares had different side lengths, like 3 inches?
2. What would happen to the volume if the original square sheet was 25 inches instead?
3. How could we maximize the volume of the box by choosing the optimal cut square size?
4. What if the volume of the box was 800 cubic inches instead of 578?
5. How does the area of the original square sheet compare to the area of the resulting box base?

**Tip**: When working with volume-related problems, always ensure that the units of each dimension (length, width, and height) are consistent to avoid calculation errors.

Suppose that an open box is to be made from a square sheet of cardboard by cutting out squares from each corner. The box is to have a volume of 578 cubic inches. Find the original dimensions of the sheet of cardboard.

Learn how to find the original dimensions of a square sheet of cardboard used to create an open box with a volume of 578 cubic inches. The problem involves cutting out squares from each corner and solving for the unknown side length using algebraic and geometric methods.

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