Finding the Number of Smaller Cubes Formed from a Cuboid

When a solid shape like a cuboid is melted and recast into smaller shapes like cubes, the total volume of the material remains the same. This means the volume of the original cuboid is equal to the combined volume of all the smaller cubes formed.

To find out how many smaller cubes can be formed, we need to calculate the volume of the cuboid and the volume of one smaller cube. Then, we can divide the volume of the cuboid by the volume of a single cube.

Calculating the Volume of the Cuboid

The dimensions of the cuboid are given as 24 cm, 9 cm, and 8 cm. The volume of a cuboid is calculated using the formula:

Volume of cuboid = Length $\times$ Width $\times$ Height

Using the given dimensions:

Volume of cuboid $= 24 \, \text{cm} \times 9 \, \text{cm} \times 8 \, \text{cm}$

Volume of cuboid $= (24 \times 9 \times 8) \, \text{cm}^3$

Volume of cuboid $= (216 \times 8) \, \text{cm}^3$

Volume of cuboid $= 1728 \, \text{cm}^3$

Calculating the Volume of a Smaller Cube

The side length of each smaller cube is given as 3 cm. The volume of a cube is calculated using the formula:

Volume of cube = Side $\times$ Side $\times$ Side = Side$^3$

Using the given side length:

Volume of smaller cube $= (3 \, \text{cm})^3$

Volume of smaller cube $= (3 \times 3 \times 3) \, \text{cm}^3$

Volume of smaller cube $= 27 \, \text{cm}^3$

Finding the Number of Cubes Formed

The total volume of the material is conserved. Therefore, the number of smaller cubes formed can be found by dividing the volume of the cuboid by the volume of one smaller cube.

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

Number of cubes = $\frac{1728 \, \text{cm}^3}{27 \, \text{cm}^3}$

Now, we perform the division:

$1728 \div 27 = 64$

So, 64 smaller cubes can be formed from the melted cuboid.

Thus, 64 smaller cubes of side 3 cm can be formed by melting a cuboid of dimensions 24 cm, 9 cm, and 8 cm.