## 214. Math Puzzle $3 / 83$

From a rectangular piece of cardboard with side lengths $a=90 \mathrm{~cm}$ and $b=45 \mathrm{~cm}$, equal-sized squares are cut out from each corner. After this, the remaining area has an area of $3650 \mathrm{~cm}^{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-71.jpg?height=329&width=650&top_left_y=1980&top_left_x=697)

a) How long is the side of each of the four cut-out squares?

b) If the resulting piece of cardboard is folded along the dashed lines, an open-top box is formed. What is the volume of the box?
The rectangular piece of cardboard has an area of $90 \times 45 \mathrm{~cm}=4050 \mathrm{~cm}^{2}$. From 4050 $\mathrm{cm}^{2}-3850 \mathrm{~cm}^{2}=400 \mathrm{~cm}^{2}$ and $400 \mathrm{~cm}^{2}: 4=100 \mathrm{~cm}^{2}$, it follows that each cut-out square has an area of $100 \mathrm{~cm}^{2}$.

The side length is $10 \mathrm{~cm}$.

Therefore, the open-top box has a volume of $(90-20) \cdot(45-20) \cdot 10=17500 \mathrm{~cm}^{3}$.