Question

Three cubes with sides in the ratio 3:4:5 are melted to form a single cube whose diognal is 12 root3cm.the sides of cubes are ​

Answer 1

Let the sides of the cubes be 3x, 4x and 5x  and that of the final cube = a

Since the cubes are melted to form the final cube

Volume of final cube = Sum of volumes of the three cubes

Therefore, a³=(3x)³ +(4x)³ +(5x)³

                a³= (27 + 64 + 125)x³

                ∛a³ = ∛216x³

                a =6x

Diagonal of a cube = a√3

for the final cube, 6x√3=12√3

                                 6x=12

                                   x=2

Therefore the sides of the cubes are 6cm, 8cm and 10cm

Answer 2

Let the side of smaller cubes be 3x,4x and 5x .

Thus, volume of bigger cube made by these cubes will be(3x)^3+(4x)^3+(5x)^3 =27x^3+64x^3+125x^3

=x^3(27+64+125)

x^3(216)

now this is the volume of bigger cube =216x^3

Let A be the side of bigger cube

this impies A^3=216x^3

therefore A=6x

Now in question it is said that diagonal of bigger cube =12roo3

we know that

Diaģonal of bigger cube =underroot3A

12root3=root3 ×6x

x=2

Now , on puttiong x=2 in 3x,4xand 5x we will get the side length of smaller cubes which is 6 unit,8units and 10 units.