Question

With no wastage, small cubes of side 3 inches are formed from a cuboid with the dimensions 2:3:9. Find the ratio of the body diagonal of the cuboid to that sum of diagonal lengths of all cubes.

• A. 1:3
• B.
• C.
• D. none of these

Answer 1

The correct option is B

Number of Cubes = $\frac{2a\times 3a\times 9a}{3\times 3\times 3}=2a^3$
Diagonal of Cuboid = $\sqrt{4a^2+9a^2+81a^2}$
Sum of the diagonals of cube = 2a${}^3$ $\times$ 3$\sqrt{3}$
Ratio = $\frac{\sqrt{4a^2+9a^2+81a^2}}{2a^3\times 3\sqrt{3}}=\frac{\sqrt{94}}{6a^2\sqrt{3}}$