Question

Three cubes of a metal whose edges are in the ratio 3 : 4 : 5 are melted and converted into a single cube whose diagonal is  `12sqrt(3)`  cm. Find the edges of the three cubes.

Answer 1

Let the edge of the metal cubes be 3x , 4x and 5x.

Let the edge of the single cube be a.

As,

Diagonal of the single cube `=12sqrt(3)  "cm"`  

`=> asqrt(3) = 12sqrt(3)`

`=> a = 12  "cm"`

Now,

Volume of the single cube = sum of the volumes of the metallic cubes 

`=> a^3 = (3x)^3 + (4x)^3 + (5x)^3`

`=> 12^3 = 27x^3+64x^3+125x^3`

`=> 1728 = 216x^3`

`=> x^3 = 1728/216`

`=> x^3 = 8`

`=> x^3 = root(3)(8)`

`=> x = sqrt(3)(8)`

`=> x = 2`

so, the edges of the given three metallic cubes are 6 cm, 8 cm and 10 cm.

Hence, the edges of the given three metallic cubes are 6 cm, 8 cm and 10 cm.