Answer:

Let the edge of cube be 3x 4x 5x

Volume =(3x) +3 (4x)+ 3 (5x) 3 =216x3

Volume of new cube =volume of old one

A3=216x3

A=6x

Digonal of new cube=12root3

Root a2+a2+a2=12root 3

Root 3a=12root3

A=12

X=2

Put the value in each

Given, edges are of the cubes are in the ratio 3 : 4 : 5.

Let these be 3k, 4k, and 5k respectively.

Their volumes are 27k³, 64k³ and 125k³ m³.

Thus,

Volume of single cube:

= (27 + 64 + 125) k³

= 216 k³

= (6k)³ m³.

Now,

We know, Length of the diagonal of a cube of side x is √3x.

Therefore the length of the single cube is equal to 6k√3.

But, length of the single cube whose diagonal is 12√3 cm..

Hence,

6k√3 = 12√3.

k = 2.

Therefore:

Length of the edges of cubes are:

6 cm, 8 cm and 10 cm.

Hope it helps!