Jerry is packing cylindrical cans with diameter 6 in. and height 10 in. tightly into a box that measures 3 ft by 2 ft by 1 ft. All rows must contain the same number of cans. The cans can touch each other. He then fills all the empty space in the box with packing foam. How many cans can Jerry pack in one box? Find the volume of packing foam he uses. What percentage of the box's volume is filled by the foam?

so each box has just 1 level of cans since height = 1ft

3ft x 2ft = 36in x 24 in
so whe have 36/6 = 6 cans in one dimension
and 24/6 = 4 cans in the other dimension
we have 6x4 = 24 cans per box ==> answer 1
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volume of the box = 36x24x12 = 10368 in^3
volume of 1 can = Pi*3²*10 = 3.1416*90 = 282.744
volume of 24 cans = 24*282.744 = 6785.856
volume of the foam = 10368-6785.856 = 3582 in^3 ==>answer 2
percentage of the box's volume is filled by the foam
= 3582/10368 = 34.6%

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.

To determine how many cans Jerry can pack in one box, we need to calculate the volume of the cans and the volume of the box.

1. Volume of a cylindrical can:
Volume = π * (radius)^2 * height = π * (diameter/2)^2 * height
Volume = π * (6/2)^2 * 10 = π * 3^2 * 10 = 90π cubic inches

2. Volume of the box:
Volume = length * width * height
Since the measurements are given in feet, we need to convert them to inches.
Volume = (3 * 12) * (2 * 12) * (1 * 12) = 36 * 24 * 12 = 10368 cubic inches

3. Number of cans that can fit in one row:
Divide the length of the box by the diameter of a can.
Number of cans = 36 * 12 / 6 = 72 cans

4. Number of rows:
Divide the width of the box by the diameter of a can.
Number of rows = 24 * 12 / 6 = 48 rows

5. Total number of cans that can fit in the box:
Multiply the number of cans in one row by the number of rows.
Total number of cans = 72 * 48 = 3456 cans

So, Jerry can pack 3456 cans in one box.

To find the volume of packing foam, subtract the volume of the cans from the volume of the box:

Volume of packing foam = Volume of the box - (Number of cans * Volume of each can)
Volume of packing foam = 10368 - (3456 * 90π)
Volume of packing foam ≈ 10368 - 279936π cubic inches

To find the percentage of the box's volume filled by the foam:

Percentage = (Volume of packing foam / Volume of the box) * 100
Percentage ≈ (10368 - 279936π) / 10368 * 100

Please note that the percentage calculation will require a calculator or a numerical approximation.

To find the number of cans Jerry can pack in one box, we need to calculate the number of cans that can fit in each row and the number of rows that can fit in the box.

First, let's determine the number of cans that can fit in each row. Since the diameter of the cans is given as 6 inches, we can calculate the length of the row as follows:

Length of the row = Width of the box = 3 ft = 36 inches

To find the number of cans that can fit in the row, we divide the length of the row by the diameter of each can:

Number of cans in each row = Length of the row / Diameter of each can
= 36 inches / 6 inches
= 6 cans

Next, let's determine the number of rows that can fit in the box. Since the height of the cans is given as 10 inches, we can calculate the height of the box that will be occupied by the cans as follows:

Height of occupied space in the box = Height of each can * Number of rows
= 10 inches * Number of rows

Since the height of the box is 1 ft, which is equal to 12 inches, we can set up the equation:

Height of occupied space in the box = Height of the box

10 inches * Number of rows = 12 inches

Solving for the number of rows:

Number of rows = 12 inches / 10 inches
= 1.2 rows

Since we need the rows to be a whole number, we round down to the nearest whole number:

Number of rows = 1 row

Therefore, Jerry can pack 6 cans in one box.

To find the volume of packing foam used, we need to subtract the volume occupied by the cans from the total volume of the box.

Volume of each can = π * (radius)^2 * height
= π * (diameter/2)^2 * height
= 3.14 * (6 in / 2)^2 * 10 in^3
= 3.14 * 3^2 * 10 in^3
= 282.6 in^3 (rounded to one decimal place)

Volume occupied by the cans = Number of cans * Volume of each can
= 6 cans * 282.6 in^3
= 1695.6 in^3 (rounded to one decimal place)

Total volume of the box = Length * Width * Height
= 36 inches * 24 inches * 12 inches
= 10368 cubic inches

Volume of packing foam used = Total volume of the box - Volume occupied by the cans
= 10368 cubic inches - 1695.6 in^3
= 8672.4 cubic inches (rounded to one decimal place)

Finally, to find the percentage of the box's volume filled by the foam, we divide the volume of packing foam used by the total volume of the box and multiply by 100:

Percentage filled by foam = (Volume of packing foam used / Total volume of the box) * 100
= (8672.4 cubic inches / 10368 cubic inches) * 100
= 83.69% (rounded to two decimal places)

Therefore, Jerry can pack 6 cans in one box, the volume of packing foam used is 8672.4 cubic inches, and the foam fills approximately 83.69% of the box's volume.