McKinney's Sweet Shop specializes in homemade candies and ice cream. McKinney's produces its ice cream in house, in batches of 50 pounds. The first stage in ice cream making is the blending of ingredients to obtain a mix, which meets pre-specified requirements on the percentages of certain constituents of the mix. The desired composition is as follows: Constituent 1. Fat 2. Serum Solids 3. Sugar Solids 4. Egg Solids 5. Stabilizer 6. Emulsifier 7. Water Required Percentage (%) 18.00 8.00 14.00 0.35 0.25 0.15 59.25 The mix can be composed of ingredients from the following list: Ingredient Cost ($/lb) 1. 40% Cream 1.13 2. 23% Cream 3. Butter 4. Plastic Cream 5. Butter Oil 0.70 2.30 2.30 2.85 6. 4% Milk 0.25 7. Skim Condensed Milk 0.42 8. Skim Milk Powder 0.65 9. Liquid Sugar 0.31 10. Sugared Frozen Fresh Egg Yolk 1.75 11. Powdered Egg Yolk 4.52 2.45 12. Stabilizer 13. Emulsifier 14. Water 1.53 0.00 The number of pounds of a constituent found in a pound of an ingredient is shown below (the rows correspond to constituents 1 through 7 and the columns correspond to ingredients 1 through 14). Note that a pound of stabilizer contributes only to the stabilizer requirement (one pound), one pound of emulsifier contributes only to the emulsifier requirement (one pound), and that water contributes only to the water requirement (one pound). Ingredient Constituent 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 0.4 0.2 0.8 0.8 0.9 0.1 0.5 0.6 2 0.1 0.1 0.1 0.3 1 3 0.7 0.1 4 0.4 0.4 5 1 6 1 7 0.5 0.8 0.2 0.1 0.1 0.8 0.7 0.3 1 Jack McKinney Jr. has recently acquired the shop from his father (Papa Jack). Jack's father has in the past used the following mixture: 9.73 pounds of Plastic Cream, 3.03 pounds of Skim Milk Powder, 11.37 pounds of Liquid Sugar, 0.44 pounds of Sugared Frozen Fresh Egg Yolk, 0.12 pounds of Stabilizer, 0.07 pounds of Emulsifier and 25.24 pounds of water. (The scale at McKinney's is only accurate to 100ths of a pound.) (a) Jack feels that perhaps it is possible to produce the ice cream in a more cost-effective manner. He would like to find the cheapest mix for producing a batch of ice cream, which meets the requirements specified above. How does the optimal cost compare to the cost of Papa Jack's recipe? Do not round intermediate calculations. Round your answer to the nearest cent. The optimized recipe saves $ per 50 pounds. (b) Jack is also curious about the cost effect of being a little more flexible in the requirements. He wants to know the cheapest mix if the composition meets the following tolerances: 1. Fat 2. Serum Solids 3. Sugar Solids 17.00-19.00% 7.00-9.00% 13.50-14.50% 0.30-0.40% 4. Egg Solids 5. Stabilizer 6. Emulsifier 7. Water 0.20-0.30% 0.10-0.20% 58.00-59.50% How much cost benefit is there to being more flexible? Do not round intermediate calculations. Round your answer to the nearest cent. The flexible recipe saves $ per 50 pounds in comparison with the optimal cost found in part (a).