To explore the packing of circles with a diameter of 24 inches in two different tiling arrangements — square and hexagonal grids — we can compare the efficiency of these two packings in terms of how well they utilize the available area.

1. Square Packing:

In a square packing arrangement, the circles are placed in a grid where each circle's center is aligned with the vertices of a square.

• Area of the circle:
  The area of each circle with diameter $d = 24$ inches can be computed using the formula:
  $A_{\text{circle}} = \pi r^2 = \pi \left( \frac{d}{2} \right)^2 = \pi (12)^2 = 144\pi \approx 452.39 \, \text{square inches}$

• Area of each square: The square’s side length is equal to the circle’s diameter (24 inches), so the area of one square is:
  $A_{\text{square}} = d^2 = 24^2 = 576 \, \text{square inches}$

• Packing efficiency in a square grid:
  The efficiency $E_{\text{square}}$ of the square packing is the ratio of the area of the circle to the area of the square:
  $E_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{144\pi}{576} \approx 0.785$ This means that about 78.5% of the area is filled with circles, while the rest is unused space.

2. Hexagonal Packing:

In hexagonal packing, the circles are arranged so that each circle is surrounded by six others, creating a tighter configuration than the square grid.

• Area of the hexagon enclosing one circle: The side length of the hexagon is equal to the radius of the circle, $r = 12$ inches. The area of a regular hexagon can be calculated using the formula: $A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 = \frac{3\sqrt{3}}{2} (12)^2 = \frac{3\sqrt{3}}{2} \times 144 \approx 374.12 \, \text{square inches}$

• Packing efficiency in a hexagonal grid:
  The efficiency $E_{\text{hexagon}}$ of the hexagonal packing is the ratio of the area of the circle to the area of the hexagon: $E_{\text{hexagon}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{144\pi}{374.12} \approx 0.815$ Thus, about 81.5% of the area is filled with circles, making this packing arrangement more efficient than the square grid.

Conclusion:

• The hexagonal packing is more efficient, with about 81.5% area utilization, compared to the square packing, which has about 78.5% area utilization.
• The hexagonal pattern allows for tighter packing of circles, making it a better option if the goal is to maximize the number of circles that can fit in a given area.

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