Understanding the Problem: Melting and Reforming Cubes

The question describes a scenario where three solid cubes, each with a different volume, are melted down and reformed into a single large cube. We are given the volumes of the original three cubes and need to find the length of the edge of the new solid cube that is formed.

The key principle here is the conservation of volume. When materials are melted and reformed into a new shape, the total volume of the material remains constant (assuming no loss during the process). Therefore, the volume of the new, larger cube will be equal to the sum of the volumes of the three original cubes.

Calculating the Edge Length of Each Original Cube

The volume of a cube is calculated using the formula: Volume = \( \text{edge}^3 \). To find the edge length of a cube given its volume, we take the cube root of the volume: edge = \( \sqrt[3]{\text{Volume}} \).

Let the edge lengths of the three original cubes be \(e_1\), \(e_2\), and \(e_3\), and their volumes be \(V_1\), \(V_2\), and \(V_3\) respectively.

• Volume of the first cube, \(V_1 = 125 \text{ cm}^3\).
• Volume of the second cube, \(V_2 = 64 \text{ cm}^3\).
• Volume of the third cube, \(V_3 = 27 \text{ cm}^3\).

Now, let's calculate the edge of each cube:

• Edge of the first cube, \(e_1 = \sqrt[3]{V_1} = \sqrt[3]{125} \text{ cm}\). Since \(5 \times 5 \times 5 = 125\), \(e_1 = 5 \text{ cm}\).
• Edge of the second cube, \(e_2 = \sqrt[3]{V_2} = \sqrt[3]{64} \text{ cm}\). Since \(4 \times 4 \times 4 = 64\), \(e_2 = 4 \text{ cm}\).
• Edge of the third cube, \(e_3 = \sqrt[3]{V_3} = \sqrt[3]{27} \text{ cm}\). Since \(3 \times 3 \times 3 = 27\), \(e_3 = 3 \text{ cm}\).

Calculating the Volume of the New Cube

The total volume of metal used to form the new cube is the sum of the volumes of the three original cubes.

\( V_{total} = V_1 + V_2 + V_3 \)

\( V_{total} = 125 \text{ cm}^3 + 64 \text{ cm}^3 + 27 \text{ cm}^3 \)

\( V_{total} = 189 \text{ cm}^3 + 27 \text{ cm}^3 \)

\( V_{total} = 216 \text{ cm}^3 \)

So, the volume of the new solid cube, \(V_{new}\), is \(216 \text{ cm}^3\).

Finding the Edge Length of the New Cube

Now that we have the volume of the new cube, we can find its edge length using the cube root formula again.

Let the edge of the new cube be \(e_{new}\).

\( e_{new} = \sqrt[3]{V_{new}} = \sqrt[3]{216} \text{ cm} \)

We need to find a number that, when multiplied by itself three times, equals 216. Let's check some small integers:

• \(1^3 = 1 \times 1 \times 1 = 1\)
• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(3^3 = 3 \times 3 \times 3 = 27\)
• \(4^3 = 4 \times 4 \times 4 = 64\)
• \(5^3 = 5 \times 5 \times 5 = 125\)
• \(6^3 = 6 \times 6 \times 6 = 216\)

Since \(6^3 = 216\), the cube root of 216 is 6.

Therefore, \(e_{new} = 6 \text{ cm}\).

The edge of the new solid cube is 6 cm.

Revision Table: Cube Calculations

Additional Information: Conservation of Volume and Mensuration

The principle of conservation of volume is fundamental in many mensuration problems involving melting and recasting solids. It states that the amount of substance, and hence its volume, remains constant regardless of its shape, assuming no loss or addition of material.

Mensuration is a branch of mathematics that deals with the measurement of lengths, areas, and volumes of geometric shapes. Understanding the formulas for basic shapes like cubes, cuboids, cylinders, cones, and spheres, and the principle of conservation of volume, is crucial for solving such problems.

• Cube: A cube is a 3D shape with six square faces, twelve edges, and eight vertices. All edges are equal in length. Volume = \( \text{edge}^3 \). Surface Area = \( 6 \times \text{edge}^2 \).
• Melting and Recasting: When a solid is melted and cast into another shape, the material is simply changing form. The total amount of material, and thus its volume, stays the same. This is why we add the volumes of the smaller shapes to find the volume of the new, larger shape.

This problem is a straightforward application of the volume formula for a cube and the conservation of volume principle.