Solve each of the following problems by the graphical/algebra method covered in class 3 (January 26th).  Show all your work. 

Each problem is identical to the corresponding problem on homework 1; the difference is that this time you should just solve the problem by hand using the graphical method, rather than on the computer.  Solutions to the algebra portion of homework 1 will be posted on Sakai by the afternoon of Monday, February 1, so you can be sure you are solving the correct formulation.  Full solutions to homework 1 will be posted after homework 2 is collected.
 

Q1. (30 points)  You operate a small brewery that makes beer and ale.  Making a barrel of beer requires 3 pounds of corn, 5 pounds of wheat, and 2 pounds of hops, while making a barrel of ale requires 4 pounds of corn, 3 pounds of wheat, and 3 pounds hops.  Your profit from making beer is $3/barrel, and your profit from making ale is $2/barrel.  If you have 60 pounds of corn, 45 pounds of wheat, and 36 pounds of hops available, how much beer and ale should you make to obtain the highest possible profit?
 

Q2. (30 points)  Problem 3 on page 20 of the course pack (the farmer with 45 acres).  Remember that the farmer may plant at most 45 total acres (assume that she do not have to plant the entire 45 acres, but can leave some land unplanted if she chooses). 
 

Q3. (40 points)  Your firm has a machine that takes large sheets of cardboard and makes them into boxes used by your shipping department.  The machine is set up to cut the sheets using two possible patterns.  When you use pattern 1, each cardboard sheet becomes 4 small boxes, 2 medium boxes, and 1 large box.  When you use pattern 2, each cardboard sheet becomes 2 small boxes, 3 medium boxes, and 2 large boxes.  This week, the shipping department needs at least 72 small boxes, at least 52 medium boxes, and at least 25 large boxes.  Each time you use pattern 1, the net cost is $4.00, and each time you use pattern 2, the net cost is $3.00.  How can you supply the shipping department's needs at the smallest possible total net cost?     (Note: this problem's feasible region looks very different from the other problems', and remember we are minimizing rather than maximizing.)