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The image asks about packing circles within squares and hexagons under two scenarios:

1. **Circles with a radius of 14 cm**.
2. **Circles with a diameter of 24 inches**.

Let's break down the problem and solution for each case.

### General Concept of Circle Packing
- **Square packing**: In a square, the circles are arranged in rows and columns. The packing efficiency (how much of the area is occupied by circles) is about 78.54% for circles in squares.
- **Hexagonal packing**: In a hexagonal arrangement, circles are packed in a honeycomb-like structure, which provides the densest packing. The efficiency here is around 90.69%.

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### 1. **Packing Circles with Radius = 14 cm**

- **Square packing**:
  - The area of a square enclosing a circle is \( (2 \times 14 \, \text{cm})^2 = 28^2 = 784 \, \text{cm}^2 \).
  - The area of one circle is \( \pi \times 14^2 = 615.75 \, \text{cm}^2 \).
  - **Packing efficiency**: \( \frac{615.75}{784} \times 100 \approx 78.54\% \).

- **Hexagonal packing**:
  - In a hexagonal arrangement, each circle touches 6 others. 
  - For hexagonal packing, the efficiency is around **90.69%**.
  - To find the area of the smallest hexagon that can contain the circle, the formula involves the side length (which is equal to the radius). The area of a regular hexagon with a side length of \( 14 \, \text{cm} \) is:
    \[
    A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times \text{side}^2 = \frac{3\sqrt{3}}{2} \times 14^2 = 509.04 \, \text{cm}^2.
    \]

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### 2. **Packing Circles with Diameter = 24 inches**
  
- **Square packing**:
  - The radius is \( \frac{24}{2} = 12 \, \text{inches} \).
  - The area of the square enclosing a circle is \( (2 \times 12 \, \text{inches})^2 = 24^2 = 576 \, \text{inches}^2 \).
  - The area of one circle is \( \pi \times 12^2 = 452.39 \, \text{inches}^2 \).
  - **Packing efficiency**: \( \frac{452.39}{576} \times 100 \approx 78.54\% \).

- **Hexagonal packing**:
  - For a circle with a radius of 12 inches in hexagonal packing, the efficiency is again around **90.69%**.
  - The area of the hexagon with a side length of \( 12 \, \text{inches} \) is:
    \[
    A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times 12^2 = 374.12 \, \text{inches}^2.
    \]

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Would you like more details on how the packing areas were computed or any other specifics? Here are 5 related questions:

1. How does the efficiency of packing change for circles of different sizes?
2. How can the total packing efficiency be improved in irregular-shaped containers?
3. What is the relationship between the radius and packing density in various polygonal shapes?
4. How does the packing arrangement affect material usage in manufacturing?
5. Can we derive a formula for packing efficiency for arbitrary polygons?

**Tip**: The hexagonal packing pattern is the most efficient for packing circles of the same size in two-dimensional space.

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

Explore the process of packing circles with a radius of 14 cm and a diameter of 24 inches into squares and hexagons. This problem covers circle packing efficiency, area formulas, and the differences between square and hexagonal arrangements.

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