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प्रश्न
Three cubes of a metal whose edges are in the ratios 3 : 4 : 5 are melted and converted into a single cube whose diagonal is \[12\sqrt{3}\]. Find the edges of the three cubes.
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उत्तर
The three cubes of metal are in the ratio 3 : 4 : 5.
Let the edges of the cubes be 3x, 4x and 5x.
Volume of the three cubes will be
\[V_1 = \left( 3x \right)^3 \]
\[ V_2 = \left( 4x \right)^3 \]
\[ V_3 = \left( 5x \right)^3\]
Diagonal of the single cube = \[12\sqrt{3} cm\]
We know diagonal of the cube = \[a\sqrt{3} = 12\sqrt{3}\]
Hence, the side of the cube = 12 cm
Volume of the bigger cube \[V_b = \left( 12 \right)^3\]
Volume of the three cubes = Volume of the single
\[\left( 3x \right)^3 + \left( 4x \right)^3 + \left( 5x \right)^3 = \left( 12 \right)^3 \]
\[ \Rightarrow 27 x^3 + 64 x^3 + 125 x^3 = 1728\]
\[ \Rightarrow 216 x^3 = 1728\]
\[ \Rightarrow x^3 = \frac{1728}{216} = 8\]
\[ \Rightarrow x = 2\]
Hence, the edges of the three cubes will be \[3 \times \left( 2 \right), 4 \times \left( 2 \right), 5 \times \left( 2 \right) = 6, 8, 10\] cm.
