Question

How many cubes can be cut out of a metre cube? Given that the parameter of the small cube is 4 cm.

Answer 1

Answer:

Step-by-step explanation:

The way I interpret this question is ‘how many cubes of length 4 cm can be made from the surface area of one cube of length 16 cm.

In that case:

Surface area of the larger cube = 16^2 * 6 = 1536 cm^2

Surface area of one of the smaller cubes = 4^2 * 6 = 96 cm^2

We want to know how many cubes, n, of area 96 cm^2 can be made from 1536 cm^2, so we have:

96n = 1536

Hence n = 16 cubes.

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This is very simple (fundamental assumption is that by cube you mean the object we normally think of as sitting in R^3, that is the thing in Euclidean 3-space that looks like a Rubik’s Cube) as what you will use is dimensional analysis. In each dimension, you can fit four “cubes”, by this I mean you can put four small cubes along one face of the big cube and they will “fill up” one of the dimensions if you put them in a row. It therefore follows that you can fit 64 cubes. You can easily verify this is a maximum, by noting that both cubes have well-defined volumes (quite obviously pavable sets), hence the maximum number of cubes without overlap will be the ratio of the volumes which is 4096 cubic centimeters/64 cubic centimers, which is 64. Therefore, we know we are done, as we have found a maximum number of cubes and we discover that we can in fact fit this number of cubes. This is by a way a rephrasing of the question of the number of cubes you can make by breaking up the bigger cube, I have approached it from the perspective of aggregating small cubes to form a large cube whereas you phrased the question to ask the number of small cubes you can get by dividing the large cube.