Question

4. What is the largest number of identical cubes whose volumes are whole numbers (1,2,3,...) that would fit in a cube

whose volume is 1080 cm3? Show all your work.

Answer 1

The largest number of identical cubes whose volumes are whole numbers that would fit in a cube of volume 1080
`cm^(3)` is 10.

Explanation:

Given that volume of larger cube is 1080
`cm^(3)`

To find number of smaller cube with volume as whole numbers can fit the larger cube:

Imagine Rubik's cube of 3x3 as larger volume and one piece of Rubik's cube is smaller cube. On one edge, three smaller cube with volume

In the same way,

Assuming one edge has 'x' number of cube with volume as whole numbers

Since, It is cube. Number of cube in other edges will be 'x' too

For smaller cube,

Take one side of smaller cube as x

Therefore, One side of larger cube will be nx

For cube, length = height = breath =nx

The volume of larger cube :
`(nx)^(3) =1080`

nx=10.2598cm

we need to find the "largest" number of cube( i.e n )

Therefore, Side x will be smallest and n will be largest.

For smallest value of x,

It is also said that " volumes are whole numbers "

smallest whole number is 1

Thus,

nx=10.2598cm

n=10.2598

n is also whole number so that n=10

The largest number of identical cubes whose volumes are whole numbers that would fit in a cube of volume 1080
`cm^(3)` is 10.