Question

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 1

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.