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The image contains two problems involving packing circles:

1. **Pack circles with a radius of 14 cm using a square and a hexagon.**
2. **Pack circles with a diameter of 24 inches using a square and a hexagon.**

### Problem 1: Packing Circles with Radius 14 cm
- **Square**:
  - The side of the square can be any value, but let's consider the minimal bounding square that fits a single circle.
  - For a square, the circle will fit exactly within the square if the square's side is equal to the circle's diameter.
  - Diameter = \(2 \times \text{Radius} = 2 \times 14 = 28\) cm.
  - Therefore, the side of the square will be 28 cm.

- **Hexagon**:
  - The circle can be inscribed in a regular hexagon, where the circle's radius is equal to the distance from the center of the hexagon to any of its vertices.
  - For a regular hexagon with side length \(a\), the radius \(R\) of the inscribed circle is equal to the side length: \(R = a\).
  - Hence, side \(a = 14\) cm.
  - The minimal enclosing hexagon will have the side length equal to the circle's radius, which is 14 cm.

### Problem 2: Packing Circles with Diameter 24 inches
- **Square**:
  - For a square, the side length will be equal to the circle’s diameter.
  - Diameter = 24 inches.
  - Hence, the side of the square will be 24 inches.

- **Hexagon**:
  - Here, the radius \(R\) of the circle is half the diameter.
  - Radius \(R = \frac{24}{2} = 12\) inches.
  - The side length \(a\) of the regular hexagon equals the radius.
  - So, \(a = 12\) inches.

### Packing Analysis
- **Square Packing**:
  - The packing density of circles in a square arrangement is \(\frac{\pi}{4} \approx 0.7854\), meaning about 78.54% of the square's area is covered by circles.
  
- **Hexagonal Packing**:
  - The packing density of circles in a hexagonal arrangement is \(\frac{\pi}{2\sqrt{3}} \approx 0.9069\), which covers about 90.69% of the hexagon's area, making it more efficient.

Would you like further details or explanations on this problem?

Here are 5 related questions:
1. How does the packing density of circles change with the shape of the container?
2. What are the advantages of hexagonal packing over square packing in various applications?
3. Can you derive the formula for the area coverage by circles in any regular polygon?
4. How would the packing change if we used triangles instead of squares or hexagons?
5. What happens if you pack circles with varying radii instead of uniform ones?

**Tip**: Remember that hexagonal packing is the most efficient way to pack equal-sized circles, as it maximizes the coverage area.

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

Explore the packing of circles with a radius of 14 cm and a diameter of 24 inches using both square and hexagonal arrangements. Understand the geometry, key formulas, and packing densities involved in this advanced problem. Suitable for students in Grades 10-12.

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