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

#### Solution

The edges of the three cubes are in the ratio 3 : 4 : 5.

So, let the edges be 3*x* cm, 4*x* cm, 5*x* cm.

The diagonal of new cube is `12sqrt(3) `cm

We need to find the edges of three cubes

Here, volume of the resulting cube,

`V = (3x)^3 + (4x)^3 + (5x)^3`

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

`= 216x^3`

Let,

l → Edge of the resulting cube

So, diagonal of the cube`= sqrt(3l)`, so

`12sqrt(3) = sqrt(3l)`

Hence,

l = 12 cm

Now;

`V=1^3`

`216x^3 = 12^3`

`(6x)^3 = 12^3`

x = 2

The edges of the three cubes are,

3x = 3× 2

= 6cm

4x = 4× 2

= 8 cm

5x = 5 × 2

= 10 cm

The edges of the three cubes are 6 cm , 8 cm and 10 cm .