To tackle the problem of arranging marbles on a flat surface within a square area defined by four lines of length 40 mm, we need to consider both the size of the marbles and the most efficient packing arrangement. Each marble has a diameter of 10 mm, which means the radius is 5 mm. The goal is to maximize the number of marbles that can fit within the given area.
Understanding the Area
The area enclosed by the four lines is a square with each side measuring 40 mm. Therefore, the total area of the square can be calculated as:
Area of the square = side × side = 40 mm × 40 mm = 1600 mm².
Marble Arrangement Options
When arranging circles (or marbles) in a confined space, two common packing methods are typically considered: square packing and hexagonal packing.
- Square Packing: In this arrangement, marbles are placed in rows and columns. Each marble occupies a square of 10 mm × 10 mm.
- Hexagonal Packing: This arrangement allows for a more efficient use of space, where each marble is surrounded by six others, leading to a denser packing.
Calculating Maximum Marbles in Square Packing
In square packing, the number of marbles that can fit along one side of the square is:
Number of marbles per side = side length / diameter = 40 mm / 10 mm = 4.
Thus, the total number of marbles that can fit in the square is:
Total marbles = 4 (rows) × 4 (columns) = 16 marbles.
Exploring Hexagonal Packing
In hexagonal packing, the arrangement is more complex but allows for more marbles to fit. The distance between the centers of two adjacent marbles in a hexagonal arrangement is equal to the diameter of the marbles, which is 10 mm. However, the vertical distance between rows is less than the diameter due to the staggered arrangement.
The vertical distance between two rows can be calculated using the formula:
Vertical distance = radius × √3 = 5 mm × √3 ≈ 8.66 mm.
Determining Rows in Hexagonal Packing
To find out how many rows can fit within the 40 mm height of the square:
Number of rows = height of square / vertical distance = 40 mm / 8.66 mm ≈ 4.6.
Since we can only have whole rows, we can fit 4 rows. Each row can fit 4 marbles, but the second row will have an additional marble due to the staggered arrangement.
Calculating Total Marbles in Hexagonal Packing
In hexagonal packing, the arrangement would look like this:
- Row 1: 4 marbles
- Row 2: 5 marbles
- Row 3: 4 marbles
- Row 4: 5 marbles
Thus, the total number of marbles in hexagonal packing is:
Total marbles = 4 + 5 + 4 + 5 = 18 marbles.
Final Thoughts on Maximum Packing
From the calculations, hexagonal packing allows for a greater number of marbles to fit within the same area compared to square packing. Therefore, the maximum number of 10 mm diameter marbles that can be arranged within a 40 mm × 40 mm square is 18 marbles.
In summary, the expression for calculating the maximum number of marbles in hexagonal packing can be generalized based on the dimensions of the area and the diameter of the marbles, leading to more efficient use of space. This approach not only maximizes the number of marbles but also illustrates the principles of geometric packing in a clear and practical manner.
