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Question
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 3x cm, 4x cm, 5x 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 .
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