a marble is 12mm diameter. What is the smallest height box that is 96mm square that can hold 200 marbles and still have the lid fit on the box without inference
Option 1-Lay 64 marbles in the bottom of the box forming an 8x8 layer.
Lay a 7x7 layer atop the bottom set where each marble in the 2nd layer nests atop 3 marbles of the lower layer. Lay the 3rd 8x8 layer atop the 2nd layer. This places 177 marbles in the box. The vertical distance between the marble centers is 9.7979mm.
The height of these three layers is 31.5958mm. With 23 marbles left to be accommodated, the box must be 41.3937mm high allowing the remaining 23 marbles to nest atop 3 marbles below.
Option 2-Lay 64 marbles in the bottom of the box forming an 8x8 layer. Lay 7 marbles atop each of the strings of 8 on the bottom, each marble contacting two marbles in the first layer. Lay another layer of 8x8 marbles atop the 2nd layer. This results in 8 rows of 184 marbles with 16 marbles left. These 16 marbles can nest in sets of 3 as in option 1. The total height required for the box becomes 6 + 10.392(2) + 9.7979 + 6 = 42.58mm.
To determine the smallest height of the box that can hold 200 marbles in a 96mm square base, while still allowing the lid to fit without interference, we first need to calculate the volume of one marble.
The volume of a sphere (marble) can be calculated using the formula:
V = (4/3) * π * r^3
Given that the diameter of the marble is 12mm, we can calculate the radius (r) by dividing the diameter by 2:
r = 12mm / 2 = 6mm
Substituting the radius value into the volume formula:
V = (4/3) * π * (6mm)^3
Now we can calculate the volume of one marble.
V = (4/3) * 3.14 * (6mm)^3
V ≈ 904.32 mm^3
Now that we know the volume of one marble, we can find the total volume required to hold 200 marbles:
Total Volume = Volume of one marble * Number of marbles
Total Volume = 904.32 mm^3 * 200
Total Volume = 180,864 mm^3
Since we want the base of the box to be 96mm square, the area of the base can be calculated by squaring 96mm:
Base Area = (96mm)^2
Base Area = 9,216 mm^2
To find the height of the box, we divide the total volume by the base area:
Height = Total Volume / Base Area
Height = 180,864 mm^3 / 9,216 mm^2
Height ≈ 19.62 mm
Therefore, the smallest height box that can hold 200 marbles in a 96mm square base, while still allowing the lid to fit without interference, is approximately 19.62mm.