Thank you for clarifying that the length of the small squares cut from the corners is \( \frac{1}{5}x \), where \( x \) is the side length of the original square sheet of cardboard.

### Updated Problem Setup:
- The original square has a side length of \( x \).
- The small squares cut from each corner have a side length of \( y = \frac{1}{5}x \).
- After cutting and folding, the side length of the box's base will be \( x - 2y \).
- Substituting \( y = \frac{1}{5}x \), the side length of the base becomes:
  \[
  x - 2\left(\frac{1}{5}x\right) = x - \frac{2}{5}x = \frac{3}{5}x
  \]
- The area of the base of the box is given as \( 84 \, \text{cm}^2 \), so:
  \[
  \left(\frac{3}{5}x\right)^2 = 84
  \]
- We need to solve this equation for \( x \).

I'll calculate this next.The calculated side length of the original square sheet of cardboard is approximately \(15.28 \, \text{cm}\).

Since the closest given option is \(15 \, \text{cm}\), option (C) seems to be the most appropriate answer.

### Would you like to go over any specific details or explore further?

Here are 5 questions related to the problem:

1. How would the box's dimensions change if the cut squares had a different proportion of the side length?
2. What happens to the area if the length of the small squares is increased?
3. How can you verify if the cut squares were correctly calculated using the given information?
4. What real-world applications use similar methods for constructing boxes from cardboard?
5. How would the problem change if the base area of the box were different?

**Tip:** When dealing with proportions like \( \frac{1}{5} \), converting them into fractions early helps simplify complex algebraic expressions.

Learn how to calculate the dimensions of a box formed from a square sheet of cardboard by cutting small squares from each corner. This problem involves geometric concepts and algebraic equations. Suitable for advanced high school students.

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