Maximum Number of 6cm Square Pieces from a 3m x 1m Rectangular Cloth

When faced with the challenge of determining the maximum number of 6cm square pieces that can be cut from a rectangular piece of cloth measuring 3m by 1m, it is crucial to approach the problem systematically. This article will walk through the process step-by-step, ensuring accurate calculations and providing insights into geometric optimization and area considerations.

Converting Units

First, it is essential to convert all dimensions to the same unit. Since the required pieces are in centimeters (cm), we convert the dimensions of the cloth from meters (m) to centimeters (cm):

Length of the cloth: 3 m 300 cm

Width of the cloth: 1 m 100 cm

Area Calculations

The next step is to calculate the area of the rectangular cloth and the area of one square piece:

Area of the rectangular cloth:

Area length times; width 300 cm times; 100 cm 30,000 cm2

Area of one square piece:

Area (side length)^2 6 cm times; 6 cm 36 cm2

To find the maximum number of 6cm square pieces that can be cut, we divide the total area of the cloth by the area of one square piece:

Number of square pieces 30,000 cm2 / 36 cm2 ≈ 833.33

Since we can only have whole pieces, we take the integer part:

Maximum number of square pieces: 833

Geometric Optimization

However, to ensure that the maximum number of pieces is indeed 833, it is also helpful to check how many pieces can fit along each dimension of the rectangle:

Along the length of 300 cm:

300 cm div; 6 cm 50 pieces

Along the width of 100 cm:

100 cm div; 6 cm ≈ 16.67 pieces → 16 pieces

Multiplying the number of pieces that can fit along each dimension:

Total pieces: 50 times; 16 800 pieces

Thus, the maximum number of 6cm square pieces that can be cut from the rectangular cloth is 800, not 833, as the practical constraint limits the number to 800.

Conclusion

In conclusion, the maximum number of 6cm square pieces that can be cut from a 3m x 1m rectangular piece of cloth is 800, based on geometric optimization and practical constraints. This method ensures no waste and maximizes the utilization of the available material.