SOLUTION: A diet is to contain at least 895 units of carbohydrates, 2290 units of proteins, and 2260 calories. Two foods are available: F1 which costs $0.02 per unit and F2, which costs $0.0
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Question 1162054: A diet is to contain at least 895 units of carbohydrates, 2290 units of proteins, and 2260 calories. Two foods are available: F1 which costs $0.02 per unit and F2, which costs $0.04 per unit. A unit of food F1 contains 1 units of carbohydrates, 4 units of proteins and 6 calories. A unit of food F2 contains 9 units of carbohydrates, 6 units of proteins and 4 calories. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimal nutrition requirements. Answer by greenestamps(13203) (Show Source):
Step 1: Graph the constraint boundary lines to determine the feasibility region, noting that the feasibility region is on or ABOVE all three constraint boundary lines.
(1)
(2)
(3)
Here is a graph of the constraint boundary lines: (1) red; (2) green; (3) blue
Step 2: Virtually every reference you find on solving this kind of problem will tell you that you need to solve pairs of equations to find the coordinates of every corner of the feasibility region and evaluate the objective function at each corner to determine the minimum cost.
That is not true. If you are new to solving problems like this, you should do that for practice.
If you don't need the practice performing that complete second step, you can determine the corner where the objective function will have its minimum value by comparing the slopes of the constraint boundary lines and the objective function.
The objective function is ; its slope is -1/2.
The objective function will have its minimum value at the corner of the feasibility region were a line with slope -1/2 will just touch the feasibility region. That slope -1/2 is between the -1/9 slope of the first constraint line and the -2/3 slope of the second; that means the minimum value of the objective function will be at the intersection of those two constraint lines.
Solving the pair of equations for constraints (1) and (2) gives an intersection point of (508,43); the objective function evaluated at that point is 508($.02)+43(.04) = $11.88.
ANSWER: The minimum cost of a diet that meets all three nutrition requirements is $11.88.
If you perform the complete standard second step, you will find that indeed the cost is greater than $11.88 at every other corner of the feasibility region.
