Final answer:
The feasible solutions to the dietician's problem can be graphed using inequalities representing the minimum requirements of additives. The corner points are found by solving these inequalities at the points where they intersect with the axes and each other. By graphing x + y ≥ 5 and 3x + 6y ≥ 18, the corner points can be identified and represent the different combinations of Food A and Food B that meet the patient's needs.
Step-by-step explanation:
To find the set of feasible solutions for a dietician planning a diet using two food combinations, Food A and Food B, we will use a graphical method. We are given that each container of Food A contains 1 unit of Additive 1 and 3 units of Additive 2, while each container of Food B contains 1 unit of Additive 1 and 6 units of Additive 2. The patient needs at least 5 units of Additive 1 and at least 18 units of Additive 2.
Let x represent the number of Food A containers and y represent the number of Food B containers. We can set up the following inequalities:
- For Additive 1: x + y ≥ 5
- For Additive 2: 3x + 6y ≥ 18
We graph these inequalities on a coordinate plane. The feasible region is the area that satisfies both inequalities and represents combinations of Food A and Food B that meet the patient's dietary requirements. The corner points of the feasible region are where the lines intersect with each other and the axes.
To find the corner points, we solve the system of inequalities:
- At x=0: 6y≥ 18, which gives us y=3. The point is (0,3).
- At y=0: 3x≥ 18, which gives us x=6. The point is (6,0).
- By solving the system of equations x+y=5 and 3x+6y=18 using substitution or elimination, we find the intersection of the two lines. This would provide us with the third corner point, if it exists within the feasible region.