$($a$)$ Determine the amount of potatoes and green beans Deena should purchase each week for the casserole to minimize the ingredient while meeting $,$ taste, and demand requirements Before she makes her final decision, Deena plans to explore the following questions independently except where otherwise indicated $)$ Deena is not very concerned about the taste of the casserole she is only concerned about meeting nutritional requirements and cutting costs. She therefore forces Alex to change the recipe to allow for at least a one to two ratio in the weight of potatoes to green beans $.$ Given the new recipe, determine the amount of potatoes and green beans Deena should purchase each week $)$ Deena decides to lower the iron requirement to $65$ mg since she determines that the other ingredients $,$ such as the onions and cream of mushroom soup $,$ also provide Iron $.$ Determine the amount of potatoes and green beans Deena should purchase each week given this new iron requirement $($d$)$ Deena learns that the wholesaler has a surplus of green beans and is therefore selling the green beans for a lower price of $$0\mathrm{.}50$ per lb: Using the same iron requirement from part $($c$)$ and the new price of green beans $,$ determine the amount of potatoes and green beans Deena should purchase each week $($e$)$ Deena decides that she wants to purchase lima beans instead of green beans since lima beans are less expensive and provide a greater amount of protein and iron than green beans. Deena again wields her absolute power and forces Alex to change the recipe to include lima beans instead of green beans. Deena knows she can purchase lima beans for $$0\mathrm{.}60$ per from the wholesaler $.$ She also knows that lima beans contain $22\mathrm{.}68$ g of protein per $10$ ounces of lima beans, $6\mathrm{.}804$ mg of iron per $10$ ounces of lima beans $,$ and no vitamin C$.$ Using the new cost and nutritional content of lima beans, determine the amount of potatoes and lima beans Deena should purchase each week to minimize the ingredient costs while meeting nutritional $,$ taste, and demand requirements $.$ The nutritional requirements include the reduced iron requirement from part $($c$).($f$)$ Will Alex be happy with the solution in part $($ e$)?$ Why or why not? $($g$)$ A healthy food advocate group meets during Body Awareness Week and determines that the Cafeteria nutritional requirements for iron are too lax and that those for vitamin C are too stringent $.$ The task force urges the cafeteria to adopt a policy that requires each serving of an entr$$e to contain at least $120$ mg of iron and at least $500$ mg of vitamin C$.$ Using potatoes and lima beans as the ingredients for the dish and using the new nutritional requirements $,$ determine the amount of potatoes and lima beans Deena should purchase each week.