A paper mill produces rolls of paper that are 10 inches wide and 100 feet long. These rolls are used for creating narrower rolls of paper that are used in cash registers, automatic teller machines (ATMs), and other devices. The narrower widths (2, 2.5, and 3 inches) needed for these devices are obtained by cutting the 10-inch rolls using pre-specified cutting patterns. Cutting pattern # 1 will cut the 10-inch roll into four rolls that are 2.5 inches each. Cutting pat-tern # 2 results in three rolls that are each 3 inches wide (leaving 1 inch of waste on the end). Cutting pattern # 3 results in one roll that is 3 inches wide and two rolls that are 3.5 inches wide. Cutting pattern # 4 results in one of the 2.5-inch rolls, one of the 3-inch rolls and one of the 3.5-inch rolls (leaving 1 inch of waste). Cutting pattern # 5 results in 1 roll that is 2.5 inches wide and two rolls that are 3.5 inches wide (leaving 0.5 inches of waste on the end). An order has been received for 2,000 of the 2.5-inch rolls, 4,000 of the 3-inch rolls, and 5,000 of the 3.5 inch rolls. How many rolls should be cut on each pattern if the company wants to minimize the total number of 10-inch rolls used? How many rolls should be cut on each pattern if the company wants to minimize the total waste?
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