Hello everything. I actually have 2 homework problems. I have already attempted, but I want to make sure it is the correct answer.

Right now, I am in college and taking finite mathematics.

First question:
A dietitian is attempting to prepare a meal from three foods, I, II, and III.
Each food contains vitamin A, protein, and vitamin C.
One unit of food I contains 3 mg of vitamin A, 20 g of protein, and 3mg of vitamin C.
One unit of food II contains 2 mg of vitamin A, 10g of protein, and 3mg of vitamin C.
One unit of food III contains 5 mg of vitamin A, 10g of protein, and 3mg of vitamin C.
The diet must consist of at least 10mg of vitamin A, 10g of protein, and 40mg of vitamin C.
Foods I, II, and III contains 5g, 2g, and 1g of fat, respectively.
If the intent of the diet is to minimize the fat content of the diet, find the number of units of each food needed in the diet.

Here is what I have done.

Let:
x1 = # of unit needed in vitamin A
x2 = # of unit needed in protein
x3 = # of unit nedded in vitamin C

So,

To minimize the fat content:

m = 5x1 + 2x2 + x3

subject to contraints are:

3x2 + 2x2 + 5x3 = 10
20x1 + 10x2 + 10x3 = 40
3x1 + 3x2 + 3x3 = 20

Dual of the contraints:

m= 10y1 + 40y2 + 20y3
3y1 + 20y2 + 3y3 = 5
2y1 + 10y2 + 3y3 = 2
5y1 + 10y2 + 3y3 = 1

introducing the slack variable:

m= 10y1 + 40y2 + 20y3
3y1 + 20y2 + 3y3 + s1 = 5
2y1 + 10y2 + 3y3 + s2 = 2
5y1 + 10y2 + 3y3 + s3 = 1

We use calucator Texas instrument ti - 83 plus, the program we use to solve these problems is called Simplex.


After I plugg all the numbers in, I got this as the answer.

M y1 y2 y3 s1 s2 s3
0 -2 1 0 0 1 0 -1 | 4
0 -3 0 0 0 1 -1 |1
0 5/3 10/3 1 0 0 1/3 | 1/3
1 70/3 80/3 0 0 0 20/3 |20/3

So,

x1 = 0
x2 = 0
x3 = 20/3

minimum of fat contect is 20/3


how can that be right?

I think it's correct. From the data you gave you can see that each of the food types delivers a fair amount of Vitamin A and protein relative to the requirement, but only a small amount of vitamin C relative to the 40 mgs you need. In other words, if you met the vitamin C requirement then you automatically satisfy the other two, regardless of the mix you choose of the 3 food types. Notice also that each food type supplies the same amount of vitamin c - so to satisfy the reqiurements you just need to have the totals of all food types add up to 20/3. Now it's simply a matter of selecting the food with the minimum fat content, and that's food III. Consequently, 20/3 of a unit of Food III is the answer - that gets you your 20 mg of vitaimin C, and more than satisfies the requirements for vitamin A and protein.