On their farm, the Friendly family grows apples that they harvest each fall and make into three productsapple butter, applesauce, and apple jelly. They sell these three items at several local grocery stores, at craft fairs in the region, and at their own Friendly Farm Pumpkin Festival for 2 weeks in October. Their three primary resources are cooking time in their kitchen, their own labor time, and the apples. They have a total of 500 cooking hours available, and it requires 3.5 hours to cook a 10-gallon batch of apple butter, 5.2 hours to cook 10 gallons of applesauce, and 2.8 hours to cook 10 gallons of jelly. A 10-gallon batch of apple butter requires 1.2 hours of labor, a batch of sauce takes 0.8 hour, and a batch of jelly requires 1.5 hours. The Friendly family has 240 hours of labor available during the fall. They produce about 6,500 apples each fall. A batch of apple butter requires 40 apples, a 10-gallon batch of applesauce requires 55 apples, and a batch of jelly requires 20 apples. After the products are canned, a batch of apple butter will generate $190 in sales revenue, a batch of applesauce will generate sales revenue of $170, and a batch of jelly will generate sales revenue of $155. The Friendlys want to know how many batches of apple butter, applesauce, and apple jelly to produce to maximize their revenues.

a) Formulate a linear programming model for this problem, and solve the model using the computer.

b) If the Friendlys were to use leftover apples to feed livestock, at an estimated cost savings worth $0.08 per apple in revenue, how would this affect the model and solution?

c) Instead of feeding the leftover apples to the livestock, the Friendlys are thinking about producing apple cider. Cider will require 1.5 hours of cooking, 0.5 hour of labor, and 60 apples per batch, and it will sell for $45 per batch. Should the Friendlys use all their apples and produce cider along with their other three products?

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