To pack circles with a diameter of 24 inches into both square and hexagonal grids, the goal is to optimize the use of space. Let's break down how each arrangement works:

1. Square Packing:

• Arrangement: Circles are placed in rows and columns, forming a square grid.
• Circle Diameter: 24 inches.
• Square Side Length: Each circle fits inside a square of side length equal to the circle's diameter, which is 24 inches.
• Packing Efficiency: The area of each square is $A_{\text{square}} = 24^2 = 576 \, \text{in}^2$.
  • The area of each circle is $A_{\text{circle}} = \pi \times \left( \frac{24}{2} \right)^2 = 452.39 \, \text{in}^2$.
  • The packing efficiency of a square grid can be computed as: $\text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} \times 100 = \frac{452.39}{576} \times 100 \approx 78.54\%$

2. Hexagonal Packing:

• Arrangement: Circles are packed in a hexagonal pattern, where each circle is surrounded by 6 other circles.
• Circle Diameter: 24 inches.
• Hexagon Side Length: In hexagonal packing, the distance between the centers of adjacent circles is equal to the diameter of the circle, which is 24 inches.
• Packing Efficiency: The area of a regular hexagon that surrounds each circle can be calculated by: $A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times \left( \frac{24}{2} \right)^2 = 748.43 \, \text{in}^2$
  • The area of each circle remains $A_{\text{circle}} = 452.39 \, \text{in}^2$.
  • The packing efficiency of a hexagonal grid is: $\text{Efficiency}_{\text{hexagonal}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} \times 100 = \frac{452.39}{748.43} \times 100 \approx 60.45\%$

Conclusion:

• The square grid provides better space utilization with an efficiency of approximately 78.54%.
• The hexagonal grid, while less efficient at 60.45%, allows for tighter packing in certain contexts due to the circular nature of the layout.

Would you like more details on the calculations or their implications? Here are a few questions that expand on this:

1. How do different packing arrangements affect practical applications like storage?
2. Can other packing methods, such as triangular grids, be considered for circle packing?
3. How does circle packing change in 3D space, such as in spheres?
4. What real-life situations require hexagonal packing instead of square packing?
5. How does the packing efficiency impact material costs or space savings?

Tip: In general, hexagonal packing is used where maximum density of circles (or spheres) is required, such as in the arrangement of atoms in certain crystal structures.