Question 541705: A packaging company is manufacturing cardboard boxes without a lid that measure 3" long, 1" wide and 1" high.
They will be cut by using a geometric net where each side of the base is simply folded down flat (Like a cross).
The boxes will be cut from sheets of cardboard that measure 1 yard by 1 yard. How many boxes can be cut from each sheet?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! If I am interpreting this problem correctly, 84 boxes can be made from a sheet of cardboard that is 1 yd by 1 yd (or more appropriately 36 inches by 36 inches).
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I see each box requiring a piece of cardboard having dimensions of 3 inches by 5 inches. A single box has a rectangular base of 1 inch by 3 inches. That base needs to be surrounded on all sides by a 1 inch wide border that can be folded up to form the sides of the box. This leads to having the 1 inch wide bottom of the box being increased by two inches for the borders (making a total of 3 inches) and the 3 inch long bottom of the box being increased by two inches for the borders (making a total of 5 inches). From this I see each box requiring a 3 by 5 inch piece of cardboard. (Each corner of this 3 inch by 5 inch piece can have 1 inch by 1 inch piece cut away and removed to allow the 1 inch borders to be folded upward to form the sides of the box.)
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Next I see cutting a 5 inch strip of cardboard that is one yard (36 inches) long. This 5 inch wide strip can be cut into twelve 3 inch wide pieces (12 pieces times 3 inches per piece from the 36 inch long strip) thereby forming 12 pieces of cardboard, each piece having dimensions of 3 inches by 5 inches.
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Next, how many of these 5 inch wide rows can be cut from from a 36 inch piece of cardboard? Dividing 5 into 36 gives an answer of 7 rows plus a 1 inch strip of cardboard left over. This 1 inch strip can be discarded, along with the four 1-inch square pieces of cardboard that are cut from each of the corners of every 3 by 5 inch piece.
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So the answer is that from a 36 inch by 36 inch piece of cardboard, you can get 7 rows that are 5 inches by 36 inches and each of these rows is long enough to cut into 12 pieces that are 3 inches wide. So the total number of boxes that can be cut from the 36 by 36 inch piece of cardboard is 7 rows times 12 per row for a total of 84 box pattern pieces per 36 by 36 inch sheet of cardboard.
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Note of caution. Each piece to make a box is 3 inches by 5 inches for a total area of 15 square inches of cardboard per box. The sheet of cardboard is 36 inches by 36 inches for an area of 1296 square inches. If you divide the area for a box (15 square inches) into the area of a sheet of cardboard (1296 square inches) you get an answer of 86.4. That implies 86 total complete boxes can be cut from the sheet. But that is 2 more boxes than the correct answer of 84. The difference comes from the 1 inch wide scrap strip. This scrap is 1 by 36 inches so it has an area of 36 square inches. If you divide that 36 square inches by the 15 square inches required for each box you get an answer of 2.4 boxes. However, it should be fairly obvious that those 2.4 boxes cannot be cut from a 1 inch wide strip. So you can't get 86.4 boxes from the sheet of cardboard. You can only get 84 boxes, but you throw away a strip that could make 2.4 more boxes if it were of the correct dimensions.
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Hope this helps you to envision how to solve this problem.
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