Question 1205327
objective function:
cholesterol = 19x + 13y which you want to minimize.
constraint inequalities:
102x + 51y >= 510
19x + 38y >= 266
x + y >= 21
x >= 0
y >= 0


graph the opposite of the constraint inequalities.
the area on the graph that is not shaded is the region of feasibility.
the minimum cholesterol levels will be at the corner points of the region of feasibility.
evaluate the objective function at each corner point to find the corner point that contains the minimum amount of cholesterol.


here's what the graph looks like.


<img src = "http://theo.x10hosting.com/2023/121001.jpg">


your corner points are (0,21) and (21,0).
the objective function is 19x + 13y.
at (0,21), the amount of cholesterol is 13 * 21 = 273.
at (21,0), the amount of cholesterol is 19 * 21 = 399.


your minimum amount of cholesterol is at (0,21)


i plotted two extra points to show you that the minimum is at the corner point.
at (10,10), the amount of cholesterol is 10 * 19 + 10 * 13 = 320 > 273.
at (20,20), the amount of cholesterol is 20 * 19 + 20 * 13 = 640 > 273.


all the constraints need to be satisfied at the minimum cost point.


at (0,21):
the amount of vitamin C = 21 * 51 = 1071 >= 510
the amount of vitamin E = 21 * 38 = 798 >= 266
total ounces of food II = 21 >= 21.


your solution is that the minimum amount of cholesterol is when you consume 21 ounces of food II and no ounces of food I.
with 21 ounces of food II, you get 51 * 21 = 1071 units of vitamin C and 38 * 21 = 798 units of vitamin E and 13 * 21 = 273 units of cholesterol.