Four identical cubes are joined end to end to form a cuboid. If the total surface area of the resulting cuboid is 648 cm²; find the length of edge of each cube.

Also, find the ratio between the surface area of the resulting cuboid and the surface area of a cube.

Given:

Total surface area of the cuboid = 648 cm²

Let a be the side of each cube.

When four identical cubes are placed adjacently, the cuboid's dimensions are:

Length = a + a + a + a = 4a

Breadth = a

Height = a

Total surface area of the cuboid = 2(lb + bh + hl)

⇒ 2(4a x a + a x a + a x 4a) = 648

⇒ 2(4a² + a² + 4a²) = 648

⇒ 2 x 9a² = 648

⇒ 18a² = 648

⇒ a² = $\dfrac{648}{18}$

⇒ a² = 36

⇒ a = $\sqrt{36}$

⇒ a = 6

Thus, the length of the edge of each cube is 6 cm.

$\text{The ratio} = \dfrac{\text{Total surface area of the cuboid}}{\text{Surface Area of one cube}}$

$= \dfrac{648}{6 \times side^2}\\[1em] = \dfrac{648}{6 \times 6^2}\\[1em] = \dfrac{648}{6 \times 36}\\[1em] = \dfrac{648}{216}\\[1em] = \dfrac{3}{1}\\[1em]$

Hence, the edge of each cube is 6 cm, and the ratio of the surface area of the resulting cuboid to that of one cube is 3:1.