Well, Jerry certainly has his hands full with all those cylindrical cans! Let's break it down, shall we?
First, let's find out how many cans Jerry can fit in one row. To do that, we need to find out the diameter of the cans in terms of feet. Since there are 12 inches in a foot, the diameter of the cans is 6/12 = 0.5 ft.
Next, let's figure out how many rows of cans can fit in the box's height of 1 ft. Well, considering the height of each can is 10 in., which is equal to 10/12 = 5/6 ft, we can place 1/(5/6) = 6/5 = 1.2 rows in the height of the box.
Moving on, let's determine how many cans can fit in the length of the box. The length of the box is 3 ft, so the number of cans in the length will be 3/0.5 = 6.
Lastly, let's compute the number of cans in the width of the box. The width of the box is 2 ft, so the number of cans in the width will be 2/0.5 = 4.
Now, let's find the total number of cans Jerry can pack in one box. Since the number of cans must be the same in each row, we take the smallest number of cans we calculated, which is 4. So, there are 4 cans in each row.
Therefore, the total number of cans Jerry can pack in one box is 4 (length) * 6 (width) * 1 (height) = 24 cans.
Now, let's move on to the volume of packing foam. The volume of the box is 3 ft * 2 ft * 1 ft = 6 ft³. Since the total volume occupied by the cans is 24 cans * π * (0.5/2)² * 10/12 = 8π ft³, the volume of the packing foam is 6 ft³ - 8π ft³.
Lastly, to find the percentage of the box's volume filled by the foam, we calculate (volume of packing foam / volume of the box) * 100. Therefore, the percentage filled by the foam is [(6 ft³ - 8π ft³) / 6 ft³] * 100.