2. The lengths of the sides of a rectangular cardboard, expressed in decimeters, are natural numbers. From each of its corners, a square with a side length of $25 \mathrm{~cm}$ has been cut out. The remaining part of the cardboard was used to form a box without a lid, whose volume is $112.5 \mathrm{dm}^{3}$. Determine all possible lengths of the sides of the original cardboard.
2. The lengths of the sides of a rectangular cardboard, expressed in decimeters, are natural numbers. A square with a side length of $25 \mathrm{~cm}$ has been cut from each of its corners. The remaining part of the cardboard was used to form a box without a lid, whose volume is $112.5 \mathrm{dm}^{3}$. Determine all possible lengths of the sides of the original cardboard.

## Solution:

![](https://cdn.mathpix.com/cropped/2024_05_30_ba00de53bdad84e80dffg-02.jpg?height=545&width=857&top_left_y=1115&top_left_x=585)

Let $a$ and $b$ be the lengths of the sides of the cardboard, and $x$ and $y$ be the lengths of those sides reduced by $50 \mathrm{~cm}$.

Assume $a > b$ as shown in the diagram.

When forming the box (a rectangular prism), one edge has a length of $25 \mathrm{~cm}$, and the lengths of the remaining edges are $x$ and $y$. The volume of the rectangular prism with edges of lengths $x$, $y$, and $25 \mathrm{~cm} (2.5 \mathrm{dm})$ is calculated using the formula $V = x \cdot y \cdot 2.5$. From this formula, we get $112.5 = x \cdot y \cdot 2.5$, which simplifies to $x \cdot y = 45$.

The possible solutions for the lengths of the edges $x$ and $y$ are given in the table.

| $x(\mathrm{dm})$ | 1 | 3 | 5 | 9 | 15 | 45 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $y(\mathrm{dm})$ | 45 | 15 | 9 | 5 | 3 | 1 |

Given the condition $x > y$ (which follows from the condition $a > b$), the possible solutions are the pairs $(9,5)$, $(15,3)$, and $(45,1)$.

The lengths of the sides $a$ and $b$ of the original cardboard are $5 \mathrm{dm}$ longer and are given in the following table:

| $a(\mathrm{dm})$ | 14 | 20 | 50 |
| :---: | :---: | :---: | :---: |
| $b(\mathrm{dm})$ | 10 | 8 | 1 |