1. Pack the circles with radius equal to 14cm using square and hexagon.
2. Pack the circles with diameter equal to 24 inches using square and hexagon.

Let's break down the two questions step-by-step. We will solve for both **square packing** and **hexagonal packing** in each scenario. Circle packing efficiency in these two arrangements is different due to the geometry of the packing method:

1. **Square packing**: Circles are arranged in rows and columns, leaving more empty space.
2. **Hexagonal packing**: Circles are arranged in a staggered pattern (hexagonal lattice), which is more space-efficient.

### Part 1: Packing circles with radius 14 cm using square and hexagon arrangements

#### Given:
- Radius of each circle, \( r = 14 \, \text{cm} \)
- Diameter of each circle, \( d = 2r = 28 \, \text{cm} \)

#### Square Packing:
- In square packing, circles are packed in rows and columns. 
- The area of one circle is \( A_{\text{circle}} = \pi r^2 = \pi \times 14^2 = 615.75 \, \text{cm}^2 \).
- The area of the square that encloses the circle is \( A_{\text{square}} = d^2 = 28^2 = 784 \, \text{cm}^2 \).

Thus, the packing efficiency for square packing is:
\[
\text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{615.75}{784} = 0.785 \, \text{or} \, 78.5\%
\]

#### Hexagonal Packing:
- In hexagonal packing, the circles are arranged in a more compact way.
- The area occupied by each circle in a hexagonal pattern is based on the area of a regular hexagon with side length equal to the radius of the circle.
- The area of this hexagon is \( A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} r^2 = \frac{3\sqrt{3}}{2} \times 14^2 = 509.19 \, \text{cm}^2 \).

Thus, the packing efficiency for hexagonal packing is:
\[
\text{Efficiency}_{\text{hexagonal}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{615.75}{509.19} = 1.21 \, \text{or} \, 121\% \text{efficiency} (\text{hexagonal packing uses less space per circle})
\]

### Part 2: Packing circles with diameter 24 inches using square and hexagonal arrangements

#### Given:
- Diameter of each circle, \( d = 24 \, \text{inches} \)
- Radius, \( r = \frac{d}{2} = 12 \, \text{inches} \)

#### Square Packing:
- The area of one circle is \( A_{\text{circle}} = \pi r^2 = \pi \times 12^2 = 452.39 \, \text{inches}^2 \).
- The area of the square that encloses the circle is \( A_{\text{square}} = d^2 = 24^2 = 576 \, \text{inches}^2 \).

Thus, the packing efficiency for square packing is:
\[
\text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{452.39}{576} = 0.785 \, \text{or} \, 78.5\%
\]

#### Hexagonal Packing:
- The area of the hexagon is \( A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} r^2 = \frac{3\sqrt{3}}{2} \times 12^2 = 436.33 \, \text{inches}^2 \).

Thus, the packing efficiency for hexagonal packing is:
\[
\text{Efficiency}_{\text{hexagonal}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{452.39}{436.33} = 1.037 \, \text{or} \, 103.7\%
\]

### Conclusion:
- For both cases, hexagonal packing is more efficient than square packing due to its compact arrangement.
- In the first case with radius 14 cm, hexagonal packing is approximately **21% more efficient**.
- In the second case with diameter 24 inches, hexagonal packing is **3.7% more efficient**.

Would you like further details or clarification?

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Here are 5 related questions:
1. How does the packing efficiency change if we reduce the circle size?
2. How do these packing methods apply to real-world applications like manufacturing?
3. What are other packing arrangements besides square and hexagonal?
4. Can packing be optimized further by varying the shapes of the container?
5. How does the boundary shape of the container affect packing efficiency?

**Tip**: When arranging objects like circles, the shape of the container can drastically affect overall efficiency, especially near the edges.

Explore the efficiency of circle packing using square and hexagonal arrangements. Compare the space efficiency for circles with a radius of 14 cm and a diameter of 24 inches. Understand the formulas and geometry behind these arrangements, including the circle area, hexagonal area, and packing efficiency.

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