Calculating Cube Side Length

The volume of a single cube is given as $39304 \text{ cm}^3$. The formula for the volume of a cube with side length '$a$' is $V = a^3$. To find the side length, we take the cube root of the volume:

$a = \sqrt[3]{39304 \text{ cm}^3}$

Calculating the cube root:

$a = 34 \text{ cm}$

Determining Cuboid Dimensions

Five identical cubes, each with side length $a = 34 \text{ cm}$, are joined end to end. This forms a cuboid with the following dimensions:

• Length ($L$): $5 \times a = 5 \times 34 \text{ cm} = 170 \text{ cm}$
• Width ($W$): $a = 34 \text{ cm}$
• Height ($H$): $a = 34 \text{ cm}$

Calculating Cuboid Lateral Surface Area

The lateral surface area (LSA) of a cuboid is the sum of the areas of its four vertical faces (excluding the top and bottom faces). The formula is:

LSA $= 2(L+W)H$

Substitute the dimensions we found:

LSA $= 2(170 \text{ cm} + 34 \text{ cm}) \times 34 \text{ cm}$

LSA $= 2(204 \text{ cm}) \times 34 \text{ cm}$

LSA $= 408 \text{ cm} \times 34 \text{ cm}$

LSA $= 13872 \text{ cm}^2$

Alternatively, using the side length '$a$': The cuboid has dimensions $5a, a, a$. The lateral surface area is $2(5a+a)a = 2(6a)a = 12a^2$.

LSA $= 12 \times (34 \text{ cm})^2 = 12 \times 1156 \text{ cm}^2 = 13872 \text{ cm}^2$.