Lata has a cuboid with dimensions 21 cm, 18 cm, and 14 cm, and we need to determine how many such cuboids are required to form a perfect cube.

Step 1: Volume of the cuboid

The volume $V_{\text{cuboid}}$ of a cuboid is given by the product of its length, width, and height: $V_{\text{cuboid}} = 21 \times 18 \times 14 = 5292 \text{ cubic cm}.$

Step 2: Finding the side length of the perfect cube

In order to form a perfect cube, the volume of the final shape must be a perfect cube number. Let the side length of the cube be $s$. The volume of the cube is: $V_{\text{cube}} = s^3.$

For the cuboids to form a perfect cube, the volume of the cube must be a multiple of the volume of the cuboid. That is, we need to find the smallest cube that can be evenly divided by the volume of the cuboid.

Step 3: Finding the least common multiple (LCM)

The dimensions of the cuboid are 21, 18, and 14. To form a cube, each dimension of the cuboid must be able to be rearranged into equal sides of the cube. Thus, we need to find the least common multiple (LCM) of the three dimensions to determine the smallest side length of the cube.

Factorizing the dimensions:

• $21 = 3 \times 7$
• $18 = 2 \times 3^2$
• $14 = 2 \times 7$

The LCM of these factors is: $\text{LCM}(21, 18, 14) = 2 \times 3^2 \times 7 = 126.$

Thus, the side length of the cube must be 126 cm.

Step 4: Volume of the cube

The volume of the cube with side length 126 cm is: $V_{\text{cube}} = 126^3 = 126 \times 126 \times 126 = 2,000,376 \text{ cubic cm}.$

Step 5: Number of cuboids

To find how many cuboids are needed, divide the volume of the cube by the volume of one cuboid: $\text{Number of cuboids} = \frac{V_{\text{cube}}}{V_{\text{cuboid}}} = \frac{2,000,376}{5292} = 378.$

Conclusion:

Lata will need 378 cuboids to form a perfect cube.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you calculate the volume of irregular shapes?
2. Can you explain how to find the least common multiple (LCM) of multiple numbers?
3. What is the difference between a cuboid and a cube in terms of volume?
4. How can you verify if a number is a perfect cube?
5. How do factorization methods help in solving geometry problems?

Tip: When working with shapes, it's useful to remember that dimensions should align properly to form a larger shape like a cube.