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Question
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 `12sqrt(3)` cm. Find the edges of the three cubes.
Solution
Let the edge of the metal cubes be 3x , 4x and 5x.
Let the edge of the single cube be a.
As,
Diagonal of the single cube `=12sqrt(3) "cm"`
`=> asqrt(3) = 12sqrt(3)`
`=> a = 12 "cm"`
Now,
Volume of the single cube = sum of the volumes of the metallic cubes
`=> a^3 = (3x)^3 + (4x)^3 + (5x)^3`
`=> 12^3 = 27x^3+64x^3+125x^3`
`=> 1728 = 216x^3`
`=> x^3 = 1728/216`
`=> x^3 = 8`
`=> x^3 = root(3)(8)`
`=> x = sqrt(3)(8)`
`=> x = 2`
so, the edges of the given three metallic cubes are 6 cm, 8 cm and 10 cm.
Hence, the edges of the given three metallic cubes are 6 cm, 8 cm and 10 cm.
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