Alice has a sheet of cardstock measuring 60 by $25 \mathrm{~cm}$. She wants to cut the sheet to build a cube, with edges measuring an integer number of centimeters. Allowing cuts but no overlapping, what is the largest volume cube she can construct?
Let $a$ be the edge of the cube that Alice wants to build. Since the lateral area of the cube measures $6 a^{2} \mathrm{~cm}^{2}$, we must have $6 a^{2} \leq 25 \times 60$, that is, $a^{2} \leq 250$ and, therefore, $a<16$. With $a=15$ we have $4=60 \div 15$ squares with a side measuring $15 \mathrm{~cm}$ and there is a remaining rectangle of 60 by $10 \mathrm{~cm}$. We can cut out a rectangle of 60 by $2.5 \mathrm{~cm}$ and the pieces marked with $\operatorname{com} \circledast$, measuring 15 by $7.5 \mathrm{~cm}$. Thus, in the figure, the dotted line indicates a fold and the solid line indicates a cut; with the two pieces of cardboard marked with $\circledast$ we form the lid.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-077.jpg?height=220&width=534&top_left_y=2077&top_left_x=841)