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The problem you've provided describes creating a box with a square base and no top from a piece of square cardboard by cutting 6-inch squares from each corner. The box's volume is given as 15,606 cubic inches, and we need to determine the size of the original square piece of cardboard.

### Step-by-Step Breakdown:

1. **Variables:**
   - Let the side length of the original square piece of cardboard be \( x \) inches.
   - Since 6-inch squares are cut from each corner, the side length of the base of the box will be \( x - 12 \) inches (subtracting 6 inches from both sides for both corners).
   - The height of the box is 6 inches, based on the squares cut from the corners.

2. **Volume of the Box:**
   The volume \( V \) of the box can be expressed as:
   \[
   V = \text{base area} \times \text{height}
   \]
   \[
   V = (x - 12)^2 \times 6
   \]
   We know that the volume of the box is 15,606 in³, so:
   \[
   (x - 12)^2 \times 6 = 15,606
   \]
   Simplifying the equation:
   \[
   (x - 12)^2 = \frac{15,606}{6} = 2,601
   \]

3. **Solve for \( x - 12 \):**
   \[
   x - 12 = \sqrt{2,601} = 51
   \]
   Therefore:
   \[
   x = 51 + 12 = 63
   \]

### Conclusion:
The side length of the original square piece of cardboard is **63 inches**.

Would you like more details on any specific part of the solution, or do you have any additional questions?

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Here are 5 questions to expand the concept:

1. How would the solution change if the squares cut from the corners were of a different size?
2. What would happen to the volume of the box if the height of the box were doubled?
3. How would you find the maximum possible volume for a box made from a fixed-size square piece of cardboard?
4. How could you calculate the surface area of the box created in this problem?
5. What is the effect on the dimensions of the box if the volume is halved?

### Tip:
For geometric optimization problems like this, always start by expressing the volume and surface area in terms of a variable. This makes it easier to apply constraints and solve for unknowns.

A box with a square base and no top is to be made from a square piece of cardboard by cutting 6 in. squares from each corner and folding up the sides. The box is to hold 15606 in³. How big a piece of cardboard is needed?

Learn how to determine the size of a square piece of cardboard needed to create a box with a square base, no top, and a volume of 15606 cubic inches. This problem involves concepts in geometry and algebra, including volume formulas and square roots. Suitable for high school students in Grades 9-12.

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