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.

### Problem 1: Packing Circles with Radius 14 cm Using Square and Hexagonal Packing

1. **Circle Radius:** \( r = 14 \) cm  
   **Circle Diameter:** \( d = 2r = 28 \) cm

#### **Square Packing**

- In square packing, circles are arranged in a grid pattern. The area of a single square in the grid is equal to the diameter of the circle squared.

- **Side length of each square:** \( d = 28 \) cm  
- **Area of one square cell:** \( A_{\text{square}} = d^2 = 28^2 = 784 \) cm\(^2\)  
- **Area of one circle:** \( A_{\text{circle}} = \pi r^2 = \pi (14)^2 = 196\pi \approx 615.75 \) cm\(^2\)  
- **Packing efficiency (square):**  
  \[
  \text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{196\pi}{784} \approx 0.785
  \]
  So, about 78.5% of the area is occupied by circles.

#### **Hexagonal Packing**

- In hexagonal packing, each circle is surrounded by 6 others, creating a honeycomb pattern that is more efficient.

- **Side length of each hexagonal cell (approximated by circle diameter):** \( d = 28 \) cm  
- **Area of one hexagonal cell:**  
  \[
  A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \left(\frac{d}{2}\right)^2 = \frac{3\sqrt{3}}{2} \left(\frac{28}{2}\right)^2 = \frac{3\sqrt{3}}{2} \times 196 \approx 509.17 \text{ cm}^2
  \]
- **Packing efficiency (hexagonal):**  
  \[
  \text{Efficiency}_{\text{hex}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{196\pi}{509.17} \approx 0.963
  \]
  So, about 96.3% of the area is occupied by circles.

### Problem 2: Packing Circles with Diameter 24 inches Using Square and Hexagonal Packing

1. **Circle Diameter:** \( d = 24 \) inches  
   **Circle Radius:** \( r = \frac{d}{2} = 12 \) inches

#### **Square Packing**

- **Side length of each square:** \( d = 24 \) inches  
- **Area of one square cell:** \( A_{\text{square}} = d^2 = 24^2 = 576 \) in\(^2\)  
- **Area of one circle:** \( A_{\text{circle}} = \pi r^2 = \pi (12)^2 = 144\pi \approx 452.39 \) in\(^2\)  
- **Packing efficiency (square):**  
  \[
  \text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{144\pi}{576} \approx 0.785
  \]
  So, about 78.5% of the area is occupied by circles.

#### **Hexagonal Packing**

- **Side length of each hexagonal cell (approximated by circle diameter):** \( d = 24 \) inches  
- **Area of one hexagonal cell:**  
  \[
  A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \left(\frac{d}{2}\right)^2 = \frac{3\sqrt{3}}{2} \left(\frac{24}{2}\right)^2 = \frac{3\sqrt{3}}{2} \times 144 \approx 373.55 \text{ in}^2
  \]
- **Packing efficiency (hexagonal):**  
  \[
  \text{Efficiency}_{\text{hex}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{144\pi}{373.55} \approx 0.963
  \]
  So, about 96.3% of the area is occupied by circles.

### Summary:
- **Square Packing Efficiency:** Approximately 78.5%
- **Hexagonal Packing Efficiency:** Approximately 96.3%

Hexagonal packing is significantly more efficient than square packing for both cases.

---

Would you like further explanations or have any questions?

### Related Questions:
1. What are other common packing patterns besides square and hexagonal?
2. How do the packing efficiencies change if the circle sizes vary?
3. Can hexagonal packing be used for 3D spheres? How does the efficiency compare?
4. What practical applications use hexagonal packing?
5. How can packing density be optimized further in manufacturing?

### Tip:
When arranging circles, hexagonal packing is generally the most space-efficient method for both small and large-scale applications.

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 the efficiency of packing circles with a radius of 14 cm and a diameter of 24 inches using square and hexagonal arrangements. Learn about the mathematical concepts of circle packing, including efficiency calculations and the geometric properties involved. This guide is ideal for students in Grades 9-12.

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