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The image contains two questions related to packing circles within different shapes (square and hexagon). Here’s the detailed analysis:

### 1. **Packing Circles with Radius Equal to 14 cm Using Square and Hexagon:**

#### **Square Packing:**
- **Circle's Diameter:** \( 2 \times \text{radius} = 2 \times 14 \, \text{cm} = 28 \, \text{cm} \)
- **Square Side Length:** For the simplest packing, if the side length of the square is equal to the diameter of the circle (28 cm), only one circle can be packed.

If the side length of the square is larger, for example, \( n \times 28 \) cm where \( n \) is an integer, then \( n \times n \) circles can be packed. 

#### **Hexagon Packing:**
- **Hexagon Side Length:** The optimal packing density for circles within a hexagon is achieved when the hexagon's side length is equal to the circle's radius.
- In a hexagonal arrangement, the packing density (ratio of the area occupied by the circles to the total area) is \( \approx 0.9069 \).

For a hexagon where the side length equals the circle's radius (14 cm), you can pack several circles in a compact arrangement. The exact number depends on the dimensions of the hexagon.

### 2. **Packing Circles with Diameter Equal to 24 Inches Using Square and Hexagon:**

#### **Square Packing:**
- **Circle's Diameter:** \( 24 \, \text{inches} \)
- **Square Side Length:** If the square side length is exactly 24 inches, only one circle fits. For larger squares, the number of circles is determined by the ratio of the square’s side length to the circle’s diameter.

#### **Hexagon Packing:**
- **Hexagon Side Length:** The most efficient packing occurs when the hexagon’s side length equals the radius of the circles (12 inches). 
- The number of circles packed depends on the hexagon size, but the packing efficiency remains high (around 90.69%).

### Summary
- **Square Packing** typically results in fewer circles packed per unit area compared to hexagonal packing.
- **Hexagonal Packing** is more efficient, especially in a large area.

Would you like to explore specific cases with different side lengths for the square or hexagon, or do you have any other questions?

### Related Questions
1. How do you calculate the area of a hexagon and a square?
2. What is the difference in efficiency between square and hexagonal packing?
3. How does the packing density change with the number of circles?
4. What other shapes could be used for efficient circle packing?
5. How can you calculate the side length of a square or hexagon that can hold a certain number of circles?

### Tip
Hexagonal packing is generally more efficient than square packing for circles, as it allows more circles to fit within a given area.

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.

Explore how to pack circles with a radius of 14cm and a diameter of 24 inches within squares and hexagons. Learn about circle packing efficiency, the formulas for area calculations, and the best geometric arrangements. Suitable for students in Grades 10-12.

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