Arrangement of Marbles in a Plane

To maximize the number of marbles that can be placed inside a square with four lines each of length 40mm, we need to find the optimal arrangement of marbles. The key to achieving this is to ensure that the marbles are packed as tightly as possible without overlapping.

Deriving the Expression for the Number of Marbles per Unit Area

To derive the expression for the number of marbles per unit area, we need to consider the geometry of the arrangement. Let's assume that the marbles are arranged in a square grid pattern, with each marble touching its neighboring marbles.

Calculating the Side Length of the Square

To calculate the side length of the square, we need to consider the diameter of the marbles and the length of the lines. Since each line has a length of 40mm, the side length of the square can be calculated as follows:

Side length of the square = Length of the line - Diameter of a marble
Side length of the square = 40mm - 10mm
Side length of the square = 30mm

Calculating the Number of Marbles per Side

Now, let's calculate the number of marbles that can be placed along each side of the square. Since the marbles are arranged in a grid pattern, the number of marbles per side can be calculated as follows:

Number of marbles per side = Side length of the square / Diameter of a marble
Number of marbles per side = 30mm / 10mm
Number of marbles per side = 3 marbles

Calculating the Total Number of Marbles

To calculate the total number of marbles that can be placed inside the square, we need to square the number of marbles per side:

Total number of marbles = (Number of marbles per side)^2
Total number of marbles = 3^2
Total number of marbles = 9 marbles

Calculating the Area

The area of the square can be calculated as follows:

Area = (Side length of the square)^2
Area = 30mm * 30mm
Area = 900mm^2

Calculating the Number of Marbles per Unit Area

Finally, we can calculate the number of marbles per unit area by dividing the total number of marbles by the area of the square:

Number of marbles per unit area = Total number of marbles / Area
Number of marbles per unit area = 9 marbles / 900mm^2
Number of marbles per unit area = 0.01 marbles/mm^2

Therefore, the expression for the number of marbles per unit area is 0.01 marbles/mm^2.

Conclusion

By arranging the marbles in a square grid pattern, with each marble touching its neighboring marbles, we can maximize the number of marbles that can be placed inside the given area. The expression for the number of marbles per unit area is derived by considering the geometry of the arrangement and dividing the total number of marbles by the area of the square.