Question

Three solid cubes have a face diagonal of `4sqrt(2)` cm each. Three other solid cubes have a face diagonal of  `8sqrt(2)` cm each. All the cubes are melted together to form a cube. The side of the cube so formed is of length ______.

Answer 1

Three solid cubes have a face diagonal of `4sqrt(2)` cm each. Three other solid cubes have a face diagonal of  `8sqrt(2)` cm each. All the cubes are melted together to form a cube. The side of the cube so formed is of length 12 cm.

Explanation:

Given: Three cubes with face diagonal `4sqrt(2)` cm each and three cubes with face diagonal `8sqrt(2)` cm each.

Step wise calculation:

1. Face diagonal d and edge a of a cube are related by `d = asqrt(2)`.

2. For the first three cubes:

`a_1 = d_1/sqrt(2)` 

= `(4sqrt(2))/sqrt(2)`

= 4 cm 

Volume of one = 4³ = 64 cm³. 

Total for three = 3 × 64 = 192 cm³.

3. For the other three cubes:

`a_2 = d_2/sqrt(2)`

= `(8sqrt(2))/sqrt(2)`

= 8 cm 

Volume of one = 8³ = 512 cm³. 

Total for three = 3 × 512 = 1536 cm³.

4. Combined volume = 192 + 1536 = 1728 cm³.

5. If the melted material forms a cube of side s, then s³ = 1728.

Since 12³ = 1728, s = 12 cm.

The side of the cube so formed is 12 cm.