how many 2.9 inch diameter spheres can fit inside a 3,071,630.6244 inches^3 container?

Question

I cant seem to figure out the answer to this problem. any help would be great

Answer 1

There is two ways to stack the balls in the crate.

One is all of the centers line up so that each ball sits exactly on top of the ball beneath. 

The total number will by how many balls side by side to fill the space.

This is given by dividing the length, width or height by the width of an individual ball.

3,071,630.6244/2.9 = 1059182 balls / side with 2.8244 inches for them to move a little.

or 1059182 * 1059182 * 1059182 =  1,188,260,812,866,976,568 balls to fill the cube

 

the second way is to place the first layer as above, the second layer at the midpoint of the four balls below.

Since the second layer would be shifted by half the ball diameter it will still fit based on remaining space left over from the above calculation. Then the height of the the second layer can be found by the pythagorean theorem.

If you can picture the three layers of balls resting in the box the height of the center of the first layer is 1/2 the diameter or 1.45 inches. Now imagine a triangle connecting the center of a ball on the first layer with the center of a ball on the second layer to the center of a ball on the third layer. The sides of the triangle are twice the radius or the diameter and the hypotenuse can be used to find the height of the two layers.

So:

A² + B² = C²

2.9² + 2.9² =C²

4.101 = c

Take half of this which is the height of the center of the second layer above the center of the bottom layer and add the radius twice to get the heoght of the two layers.

so:

4.101/2 +1.45 *2 = 4.951

now use this to find how many two layers can fit in the height.

3,071,630.6244/4.951 = 620406 (two layers of balls fit within each other)

so the total number stacked this way is

1059182 * 1059182 = 1121866509124 per layer

(1121866509124 per layer *2 layers) * 620406 = 1,392,025,426,919,168,688 balls

1,188,260,812,866,976,568 balls stacked atop each other or

1,392,025,426,919,168,688 balls stacked nestled 

as you can see the most efficient use of space is by stacking them to fit 17% more.

Answer 2

Number of spheres = Volume of container / volume of one sphere