A marble is 12mm in diameter what is the smallest height of a box that is 96mm square that can hold 200 marbles and still have the lid fit on without interference?
If the marbles are packed in a square grid, the box will hold 8x8=64 marbles in each layer.
So, since it will take 4 layers to hold all 200 marbles, the box must be 48mm high.
If you want to get into most efficient packing, that is a complex subject, but you can research it to find that more than 64 marbles can fit in each layer, and the layers can pack more tightly than 12mm apart.
Well, Steve sorry to steal your thunder, but it would actually be 4.8 cm if it was 48mm high. Because you would have t convert itinto centimeters because the answer for the question requires that it be in centimeters. Your welcome Valina, I had kinda got lost for a moment too.
-Tea
To find the smallest height of a box that can hold 200 marbles without interference, we need to consider the diameter of the marbles and the dimensions of the box.
First, let's find the volume of one marble. The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius. Since the diameter of the marble is 12mm, the radius is half of that, which is 6mm.
V = (4/3)π(6mm)^3
≈ 904.78 mm^3
Next, let's find the total volume required to hold 200 marbles. Since we have 200 marbles, the total volume required will be:
Total Volume = V * number of marbles
= 904.78 mm^3 * 200
= 180,956 mm^3
Now, let's consider the dimensions of the box. Since the box is square and has a height, we can calculate the volume of the box by multiplying the area of the base (96 mm x 96 mm) by the height (h).
Volume of the box = Base area * height
= (96mm x 96mm) * h
= 9216 mm^2 * h
We want the total volume required to hold the 200 marbles to fit within the volume of the box without any interference. Therefore, we can set up an equation to find the minimum height (h) of the box.
9216 mm^2 * h = 180,956 mm^3
Now we can solve for h:
h = 180,956 mm^3 / 9216 mm^2
≈ 19.64 mm
Therefore, the smallest height of a box that is 96mm square and can hold 200 marbles without interference is approximately 19.64 mm.