The volume of a cuboid whose sides are in the ratio 1 : 2 : 4 is same

1.	The volume of a cuboid whose sides are in the ratio 1 : 2 : 4 is same as that of a cube. What is the ratio of the length of diagonal of the cuboid to that of the cube ?
Answer» ong>Answer: $\huge\underline\bold {Answer:}$ Let the sides of the CUBOID be x, 2x and 4x units. Let the length of each EDGE of the cube = a units. Then, a^3 = (x) (2x) (4x) => a^3 = 8x^3 => a = 2x. Length of DIAGONAL of the cuboid $= \sqrt{ {x}^{2} + (2x) {}^{2} + (4x {}^{2} ) }$ $= \sqrt{ {x}^{2} + 4 {x}^{2} + 16 {x}^{2} } \\ = \sqrt{21x {}^{2} }$ Length of diagonal of the cube $= a \sqrt{3} \\ = 2x \sqrt[]{3} \\ = \sqrt{12 {x}^{2} }$ THEREFORE, required RATIO $= \frac{ \sqrt{21 {x}^{2} } }{ \sqrt{12 {x}^{2} } } \\ = \frac{ \sqrt{21} }{ \sqrt{21} } \\ = \sqrt{1.75}$ Hence ratio of the length of the diagonal of the cuboid to that of the cube.