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 $12 \sqrt{3}$ cm. Find the edges of the three cubes.

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Solution

Let the edges of three cubes are 3x,4x and 5x respectively.

Volume of three subes = $(3 x)^{3} + (4 x)^{3} + (5 x)^{3} = 216 x^{3}$

Let a be the edge of the new cube formed.

Now Volume of new cube = volume of three cubes

=> $a^{3} = 216 x^{3}$

=> $a = 6 x$

Given, diagonal of new cube = $12 \sqrt{3}$

=> $\sqrt{3 a} = 12 \sqrt{3}$

=> a = 12

=> x = 2

Edges of cubes are 3x = 6 cm, 4x = 8 cm , 5x = 10 cm

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