QUIZ 4
Name
Prob. 1 (Set up, but do NOT solve the following linear programming problem.) A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. She has decided that the meals should contain a minimum of 400 mg of calcium, 10 mg of iron, and 40 mg of vitamin C. She has further decided that the meals are to be prepared from foods A and B. Each ounce of food A contains 30 mg of calcium, 1 mg of iron, 2 mg of vitamin C, and 2 mg of cholesterol. Each ounce of food B contains 25 mg of calcium, 0.5 mg of iron, 5 mg of vitamin C, and 5 mg of cholesterol. Find how many ounces of each type of food should be used in a meal so that the cholesterol content is minimized and the minimum requirements of calcium, iron, and vitamin C are met. (DO NOT SOLVE.)
We begin by defining our variables
Next, we know that we want to minimize chlesterol intake, and
that each ounce of Food A has 2 mg of cholesterol while each ounce of
Food B has 5mg of cholesterol. Thus we have the equation:
2x+5y=C
This is to be minimized subject to,
Prob. 2 Minimize the function C= 2x + 5y subject to
We want to graph the lines:
4x+y+40,
2x+y+30 and
x+3y=30.
The region we want is the unbounded region in the first
quadrant above all the lines. The corner points are the points
(0,40), (30,0), (5,20) and (12,6). Plugging these values into
our equation for C we get the numbers 200, 60, 110 and 54. Since 54
is the smallest and we got that number from the point (12,6) we know
that the minimum amount of cholesterol will be 54 and that we obtain
this amount when we serve 12 ounces of Food A and 6 ounces of Food B.