Six spherical cannon balls are tightly packed into a rectangular box in one layer. Each row has two cannon balls and each column has three. What part of the box is empty?Choices:- 15/21 14/21 10/21 14/23

To solve this problem, we need to understand that when spheres are tightly packed, they occupy approximately 74% (or 0.74) of the space. This is known as the packing efficiency of spheres.

1. First, we calculate the total volume of the box. Since the box contains 2 rows and 3 columns of cannon balls, the dimensions of the box are 2d x 3d x d, where d is the diameter of a cannon ball. So, the volume of the box is V_box = 2d * 3d * d = 6d^3.

2. Next, we calculate the total volume of the 6 cannon balls. The volume of a sphere is given by the formula V_sphere = 4/3 * π * (d/2)^3. So, the total volume of the 6 cannon balls is V_balls = 6 * (4/3 * π * (d/2)^3) = 8πd^3/3.

3. The packing efficiency of spheres tells us that the cannon balls occupy about 74% of the box, or 0.74 * V_box = 0.74 * 6d^3 = 4.44d^3.

4. To find the part of the box that is empty, we subtract the volume occupied by the cannon balls from the total volume of the box: V_empty = V_box - V_balls = 6d^3 - 4.44d^3 = 1.56d^3.

5. Finally, we express this volume as a fraction of the total volume of the box: 1.56d^3 / 6d^3 = 0.26, or 26%.

So, approximately 26% of the box is empty. However, none of the given choices exactly match this result. The closest choice is 10/21, which is approximately 0.476 or 47.6%. This is much higher than our calculated result, so it seems there may be an error in the problem or the given choices.