A) \[\sqrt{1.25}\]
B) \[\sqrt{1.75}\]
C) \[\sqrt{2}\]
D) \[\sqrt{3.5}\]
Correct Answer: B
Solution :
Let the sides of the cuboid are x, 2x and 4x and the side of the cube is y. \[\therefore \] Volume of cuboid \[=x\times 2x\times 4x=8{{x}^{3}}\] and volume of cube \[={{y}^{3}}\] By given condition, Volume of cuboid = Volume of cube \[\therefore \] \[8{{x}^{3}}={{y}^{3}}\]\[\Rightarrow \]\[{{\left( \frac{x}{y} \right)}^{3}}={{\left( \frac{1}{2} \right)}^{3}}\] \[\Rightarrow \] \[\frac{x}{y}=\frac{1}{2}\] \[\Rightarrow \] \[y=2x\] ?(i) \[\therefore \] Diagonal of cuboid \[=\sqrt{{{x}^{2}}+4{{x}^{2}}+16{{x}^{2}}}=\sqrt{21}x\] and diagonal of cube \[=y\sqrt{3}=2x\sqrt{3}\] [from Eq. (i)] Hence, required ratio \[=\frac{\sqrt{21}x}{2x\sqrt{3}}=\sqrt{\frac{21}{4\times 3}}=\sqrt{1.75}\]
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