Question

12. 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 12√3 cm. Find the edges of the three cubes.
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Answer 1

Answer :-

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

Formulae used :-

• Volume of cube = $\sf{{(side)}^{3}}$

• Diagonal of cube = $\sf{\sqrt{3} a}$

Step - by - Step Explanation :-

Refer to the attachment.

In the question it is given that the 3 cubes whose ratio of edges is given are melted and recasted to form a single cube.

This single cube has a diagonal of 12√3 cm.

We know that,

volume of 3 cubes = volume of single cube formed from these cubes.

Answer 2

GIVEN :

Three cubes are melted and made into single cube.

Diagonal of the large cube = 12√3cm

Ratio of edges of three cubes = 3:4:5

Let the each edge of the cube be x

Edges of three cubes be 3x, 4x and 5x

Volume of 3 cubes = Volume of new cube

We know that,

Volume of Cube = a³

Volume of three cubes :

= (3x)³ + (4x)³ + (5x)³

= 27x³ + 64x³ + 125x³

= 216x³cm

Diagonal of new cube = 12√3cm

We know that,

Diagonal of a cube = Edge√3

Side√3 = 12√3/√3

Side = 12

Side of new cube = 12

Volume of new cube = (12)³ = 1728

According to the problem,

216x³ = 1728

x³ = 1728/216

x³ = 8

x = 3√8

x = 2

Substitute value of x in the place of x.

EDGES :

3x = 3(2) = 6cm

4x = 4(2) = 8cm

5x = 5(2) = 10cm

Therefore, the edges are 6cm, 8cm and 10cm.