Problem 3

A box whose shape is a rectangular parallelepiped can be completely filled with cubes of side

1

. If one places inside it the maximum possible number of cubes, each of volume

2

, with their sides parallel to those of the box, then exactly

40\%

of the volume of the box is occupied. Determine the possible dimensions of all such boxes.

Step 2 of 6: Translate the

40\%

condition into a ratio equation

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In plain words

Dividing the volume condition through by $xyz$ turns a statement about two volumes into a statement about the product of three per-edge ratios, each close to $\sqrt[3]2$ — this is what makes the problem tractable edge by edge.

\frac{a}{x}\cdot\frac{b}{y}\cdot\frac{c}{z}=5

Detailed analysis

The hypothesis $V_2=0.4\,V_1$ reads $2xyz=0.4\,abc$ , i.e. $abc=5xyz$ , which rearranges to $\dfrac ax\cdot\dfrac by\cdot\dfrac cz=5$ .