Understanding the Problem: Cuboid Melted into Cubes

The question involves a metallic cuboid that is melted and then reformed into smaller cubes. This is a classic problem in Mensuration where the principle of volume conservation is applied. When a solid is melted and recast into a different shape, its total volume remains the same, assuming no material is lost.

We are given the dimensions of the cuboid and the side length of the smaller cubes. We need to find out how many such smaller cubes can be formed from the material of the cuboid.

Calculating the Volume of the Cuboid

The volume of a cuboid is calculated by multiplying its length, width, and height.

Dimensions of the cuboid are 36 cm × 18 cm × 12 cm.

Volume of cuboid $\text{V}_{\text{cuboid}}$ = length $\times$ width $\times$ height

$\text{V}_{\text{cuboid}} = 36 \text{ cm} \times 18 \text{ cm} \times 12 \text{ cm}$

$\text{V}_{\text{cuboid}} = 648 \text{ cm}^2 \times 12 \text{ cm}$

$\text{V}_{\text{cuboid}} = 7776 \text{ cm}^3$

Calculating the Volume of One Cube

The volume of a cube is calculated by cubing its side length.

Side of the cube is 6 cm.

Volume of one cube $\text{V}_{\text{cube}}$ = side $\times$ side $\times$ side = side$^3$

$\text{V}_{\text{cube}} = (6 \text{ cm})^3$

$\text{V}_{\text{cube}} = 6 \text{ cm} \times 6 \text{ cm} \times 6 \text{ cm}$

$\text{V}_{\text{cube}} = 36 \text{ cm}^2 \times 6 \text{ cm}$

$\text{V}_{\text{cube}} = 216 \text{ cm}^3$

Finding the Number of Cubes Formed

Since the total volume of the material is conserved, the volume of the cuboid is equal to the sum of the volumes of all the small cubes formed. If 'n' is the number of cubes formed, then:

Volume of cuboid = n $\times$ Volume of one cube

So, the number of cubes formed 'n' can be found by dividing the volume of the cuboid by the volume of one cube.

Number of cubes n $= \frac{\text{Volume of cuboid}}{\text{Volume of one cube}}$

$n = \frac{7776 \text{ cm}^3}{216 \text{ cm}^3}$

Let's perform the division:

$n = \frac{7776}{216}$

We can simplify this fraction. Both numbers are divisible by common factors. Let's start by dividing both by 6:

$7776 \div 6 = 1296$

$216 \div 6 = 36$

So, $n = \frac{1296}{36}$

Again, both are divisible by 6:

$1296 \div 6 = 216$

$36 \div 6 = 6$

So, $n = \frac{216}{6}$

Finally, perform the division:

$n = 36$

Thus, 36 cubes of side 6 cm can be formed from the metallic cuboid.

Result Summary

The volume of the original cuboid is 7776 cm$^3$. The volume of each smaller cube is 216 cm$^3$. By dividing the total volume by the volume of a single cube, we find the number of cubes formed.

Number of cubes = $\frac{7776}{216} = 36$

The number of cubes formed is 36.

Revision Table: Volume and Recasting Concepts

Additional Information: Solids and Their Volumes

Understanding the volumes of common 3D shapes is fundamental in Mensuration. Here are some basic shapes and their volume formulas:

• Cuboid: A rectangular solid with 6 faces. Volume = length $\times$ width $\times$ height.
• Cube: A special type of cuboid where all sides are equal. Volume = side$^3$.
• Cylinder: A shape with two parallel circular bases and a curved surface. Volume = $\pi r^2 h$, where r is the radius of the base and h is the height.
• Cone: A shape with a circular base and a vertex. Volume = $\frac{1}{3} \pi r^2 h$, where r is the radius of the base and h is the height.
• Sphere: A perfectly round 3D object. Volume = $\frac{4}{3} \pi r^3$, where r is the radius.

Problems involving melting and recasting often apply the principle of volume conservation. This means the total amount of material (and thus volume) before reshaping is equal to the total volume after reshaping, provided no material is added or removed.