To solve the problem of packing circles with a diameter of 24 inches into square and hexagonal arrangements, let's first look at both arrangements separately.

1. Square Packing Arrangement

In a square packing arrangement, circles are arranged in a grid where each circle is directly above and beside another circle. The distance between the centers of adjacent circles is equal to the diameter of the circles (24 inches in this case).

For a square packing arrangement, each circle occupies a square with a side length equal to its diameter. The area of the square around each circle is:

$A_{\text{square}} = \text{diameter}^2 = 24^2 = 576 \text{ square inches.}$

The area of each circle is:

$A_{\text{circle}} = \pi \left( \frac{\text{diameter}}{2} \right)^2 = \pi \left( \frac{24}{2} \right)^2 = \pi \times 12^2 = 144\pi \text{ square inches.}$

To find the packing efficiency of the square arrangement (i.e., the ratio of the circle's area to the square area it occupies):

$\text{Packing efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{144\pi}{576} = \frac{\pi}{4} \approx 0.785.$

This means approximately 78.5% of the space is used by the circles, and the rest is empty space.

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2. Hexagonal Packing Arrangement

In a hexagonal packing arrangement, circles are arranged such that each circle is surrounded by six others. This creates a denser arrangement compared to square packing. In this arrangement, the distance between the centers of two adjacent circles is still the diameter, but now each circle fits more closely with its neighbors.

For hexagonal packing, the area occupied by each circle corresponds to a regular hexagon with a side length equal to the radius of the circle (12 inches). The area of a regular hexagon with side length $s$ is given by:

$A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 = \frac{3\sqrt{3}}{2} \times 12^2 = 432\sqrt{3} \text{ square inches.}$

The packing efficiency for the hexagonal arrangement is higher than the square arrangement. It's given by:

$\text{Packing efficiency}_{\text{hexagonal}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}}.$

Theoretical calculations show that the packing efficiency for a hexagonal arrangement is approximately 0.9069, meaning about 90.69% of the space is occupied by circles, making this the denser arrangement.

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Summary

• Square Packing: Packing efficiency ≈ 78.5%.
• Hexagonal Packing: Packing efficiency ≈ 90.7%.

The hexagonal arrangement packs the circles more efficiently, leaving less wasted space.

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Do you want more details or have any questions?

Here are 5 follow-up questions:

1. How many circles can be packed in a given area using square and hexagonal arrangements?
2. How does packing efficiency change with different circle diameters?
3. What are some real-world applications of packing problems?
4. Can you visualize the difference between square and hexagonal packing patterns?
5. How would the packing efficiency be affected by irregular circle sizes?

Tip: Hexagonal packing is often found in nature, such as in honeycomb structures, because it optimizes space usage.