John owns a metal smelting factory that produces a special type of metal rod. These rods are made from titanium and are always 9 inches in diameter and 110 inches long. However, John's customers do not want the entire 110 inches of metal. Some clients only need 12 inches, others needs 15 inches, and some need only 30 inches. Therefore, John has to cut his 110 inch long metal rod into smaller pieces to satisfy his customer's demand. However, this is not a trivial task. Suppose a client needs three 12 inch rods and four 15 inch rods. John does not simply want cut his 110 inch rod into these partitions since he will have 14 inches of scrap which goes to waste (12+12+12+ 15 + 15 + 15 + 15 = 96). Therefore, he has created pre-determined cutting designs to attempt to minimize waste. John's fabrication team has five pre-determined cutting designs as shown in the table below. Suppose next month his customers demand a total are 600 - 12 inch rods, 775 - 15 inch rods, 610 30 inch rods. John should optimally select the designs that satisfy the demands while minimizing waste, otherwise he will have to spend time and labor melting the metal, again. Cut sizes Design 12 inches 15 inches 30 inches Waste 1 0 7 0 5 inches 2 1 0 3 8 inches 3 9 0 0 2 inches 4 2 1 2 11 inches 5 7 1 0 11 inches a. (6 pts) Develop an algebraic model for this problem. b. (6 pts) Solve this problem in Excel. c. (3 pts) How many rods will each design produce? d. (3 pts) What is the total wasted metal? e. (3 pts) You are tasked with creating a 6th design to further minimize total waste. What are the specifications of your design? Resolve this problem with your 6th design included. What is the total wasted metal?