Exercise 4.* Suppose that exactly four foods are available to you: milk, chocolate chip cookies, chicken soup and Brussels sprouts. Your daily diet at the moment consists of 2 quarts of milk, 2 dozen cookies, 3 cups of soup and pound of Brussels sprouts. (You actually despise Brussels sprouts, but include them because they are "good for you".) You have recently been informed that you must reduce your daily intake of fat. You wish to find a new diet that maximizes gastronomical enjoyment while also satisfying the following two conditions: (1) the daily fat intake must be reduced by at least 3000 units compared to the amount consumed now; (2) you must consume at least 600 units of vitamin X, 300 units of vitamin Y, and 550 units of vitamin Z each day. The relevant measurements for the four foods are shown, where the numerical entries denote units: Food X Y Z Fat Enjoyment Milk (quart) 50 10 150 800 200 Cookies (dozen) 3 10 35 6000 6000 Soup (cup) 150 75 75 1000 3000 Brussel sprouts (pound) 100 100 5 400 -200 Formulate a linear program whose solution will tell you the "best" new diet. Write the problem in the form max cx subject to Ax > b. Do not attempt to solve the LP. Startup matlab and enter the quantities c, A and b. When you are sure that you have the problem stated correctly, type save diet. This will save all variables that are currently in your workspace in a file diet.mat. On any subsequent occasion, you can reload the diet problem by typing the command load diet. You will need the diet problem for the next exercise. Exercise 5.* You have formulated the diet problem above as a linear program of the form max er subject to Ar > b. (a) In class we defined a "corner point" as a feasible point that lies at the intersection of n hyperplanes. Give an upper limit on the number of corner points for the diet problem. (Don't just guess a number, give an estimate based on the row and column dimensions of the constraint matrix.)