You are given marbles of diameter 10mm. They are to be placed such that their centres are lying in a square bound by four lines each of length 40mm. What will be the arrangements of the marbles in a plane so that maximum number of marbles can be placed inside the area? calculate the no. of sphere per unit area.

To determine how many marbles can fit within a square area of 40mm by 40mm, we need to consider the size of the marbles and how they can be arranged efficiently. Each marble has a diameter of 10mm, which means the radius is 5mm. The goal is to maximize the number of marbles placed within the square while ensuring that their centers remain within the boundaries of the square.

Understanding the Arrangement

Since the marbles are spherical, their centers must be spaced apart to avoid overlap. The centers of the marbles need to be at least 10mm apart (the diameter) to ensure they do not touch each other. Given the square's dimensions, we can visualize how to arrange the marbles in a grid pattern.

Calculating the Number of Marbles

Let's break down the arrangement step by step:

• The square has a side length of 40mm.
• Each marble occupies a space of 10mm in diameter.
• To find out how many marbles can fit along one side of the square, we divide the side length by the diameter of the marbles:

Number of marbles per side:

Number of marbles = Side length / Diameter = 40mm / 10mm = 4

This means we can fit 4 marbles along each side of the square.

Calculating the Total Number of Marbles

Since the arrangement is a grid, the total number of marbles that can fit in the square is the square of the number of marbles per side:

Total number of marbles:

Total marbles = Number of marbles per side × Number of marbles per side = 4 × 4 = 16

Determining the Sphere per Unit Area

Next, we need to calculate the number of marbles per unit area. The area of the square is:

Area of the square:

Area = Side length × Side length = 40mm × 40mm = 1600mm²

Now, we can find the number of marbles per unit area:

Marbles per unit area:

Marbles per unit area = Total number of marbles / Area = 16 / 1600mm² = 0.01 marbles/mm²

Summary of Findings

In summary, you can fit a maximum of 16 marbles within a 40mm by 40mm square when arranged in a grid pattern. This results in a density of 0.01 marbles per square millimeter. This arrangement is efficient and ensures that the marbles do not overlap while maximizing the use of space within the given area.