Understanding Cuboids and Cubes: Calculating Surface Area

This problem involves the concept of volume conservation when a solid object is melted and reshaped. A solid metallic cuboid is melted and recast into several smaller, identical cubes. We need to find the sum of the lateral surface areas of two of these cubes.

Key Concepts

• Volume of a Cuboid: The volume of a cuboid with length \(l\), width \(w\), and height \(h\) is given by \(V_{cuboid} = l \times w \times h\).
• Volume of a Cube: The volume of a cube with side length \(a\) is given by \(V_{cube} = a^3\).
• Lateral Surface Area of a Cube: The lateral surface area (LSA) of a cube is the sum of the areas of its four vertical faces. For a cube with side \(a\), LSA \( = 4a^2\).
• Volume Conservation: When a solid is melted and reshaped, its volume remains constant.

Step-by-Step Solution for Calculating Surface Area

1. Calculate the Volume of the Cuboid

The dimensions of the solid metallic cuboid are given as 12 cm, 54 cm, and 72 cm.

Volume of cuboid \( = l \times w \times h \)

Volume of cuboid \( = 12 \text{ cm} \times 54 \text{ cm} \times 72 \text{ cm}\)

Let's calculate this volume:

\(V_{cuboid} = 12 \times 54 \times 72\)

We can find the prime factorization to make calculation easier:

• \(12 = 2^2 \times 3\)
• \(54 = 2 \times 3^3\)
• \(72 = 8 \times 9 = 2^3 \times 3^2\)

\(V_{cuboid} = (2^2 \times 3) \times (2 \times 3^3) \times (2^3 \times 3^2)\)

\(V_{cuboid} = 2^{(2+1+3)} \times 3^{(1+3+2)} = 2^6 \times 3^6\)

\(V_{cuboid} = (2 \times 3)^6 = 6^6 \text{ cm}^3\)

\(6^6 = (6^3)^2 = (216)^2 = 46656\). Or \(6^6 = 6 \times 6 \times 6 \times 6 \times 6 \times 6 = 36 \times 36 \times 36 = 1296 \times 36 = 46656\).

So, the volume of the cuboid is 46656 cm³.

2. Determine the Volume and Side Length of Each Cube

The cuboid is melted and converted into 8 cubes of the same size. This means the total volume is divided equally among the 8 cubes.

Volume of 8 cubes = Volume of cuboid

Volume of each cube \( = \frac{\text{Volume of cuboid}}{8} \)

\(V_{cube} = \frac{6^6}{8} = \frac{(2 \times 3)^6}{2^3} = \frac{2^6 \times 3^6}{2^3}\)

\(V_{cube} = 2^{(6-3)} \times 3^6 = 2^3 \times 3^6\)

Let the side length of each cube be \(a\). The volume of a cube is \(a^3\).

\(a^3 = 2^3 \times 3^6\)

To find \(a\), we take the cube root of both sides:

\(a = \sqrt[3]{2^3 \times 3^6}\)

\(a = (2^3)^{1/3} \times (3^6)^{1/3}\)

\(a = 2^{3 \times \frac{1}{3}} \times 3^{6 \times \frac{1}{3}}\)

\(a = 2^1 \times 3^2\)

\(a = 2 \times 9\)

\(a = 18 \text{ cm}\)

The side length of each cube is 18 cm.

3. Calculate the Lateral Surface Area of One Cube

The lateral surface area (LSA) of a cube is the area of its four side faces (excluding the top and bottom faces). Each face is a square with side length \(a\).

Area of one face \( = a^2 \)

LSA of one cube \( = 4 \times \text{Area of one face} = 4a^2 \)

Using the side length \(a = 18 \text{ cm}\):

LSA of one cube \( = 4 \times (18 \text{ cm})^2 \)

LSA of one cube \( = 4 \times (18 \times 18) \text{ cm}^2 \)

\(18 \times 18 = 324\)

LSA of one cube \( = 4 \times 324 \text{ cm}^2 \)

\(4 \times 324 = 1296\)

The lateral surface area of one cube is 1296 cm².

4. Calculate the Sum of Lateral Surface Areas of Two Cubes

We need to find the sum of the lateral surface areas of 2 such cubes.

Sum of LSA of 2 cubes \( = 2 \times (\text{LSA of one cube}) \)

Sum of LSA of 2 cubes \( = 2 \times 1296 \text{ cm}^2 \)

\(2 \times 1296 = 2592\)

The sum of the lateral surface areas of 2 such cubes is 2592 cm².

Let's summarise the dimensions and calculated values in a table:

The sum of the lateral surface areas of 2 such cubes is 2592 cm².

Revision Table: Cuboid and Cube Concepts

Additional Information: Melting and Reshaping Solids

When a solid object like a cuboid is melted and recast into another shape, say multiple small cubes, the following properties are important:

• Volume: The total volume of the material remains constant throughout the process. This is a fundamental principle used in such problems.
• Surface Area: The surface area usually changes. Melting a large shape into smaller shapes typically increases the total surface area. For example, melting a large cube into 8 smaller cubes of equal size increases the total surface area.
• Material Density: The density of the material remains the same, assuming no material is lost or added during the process. Density is mass per unit volume. Since volume is conserved and material is conserved (mass is conserved), density also remains constant.

Understanding the conservation of volume is crucial for solving problems involving melting and reshaping of solids.