Please Help! One gram of soybean meal provides at least 2.5 units of vitamins an

Question

Please Help!

One gram of soybean meal provides at least 2.5 units of vitamins and 5 calories. One gram of meat byproducts provides at least 4.5 units of vitamins and 3 calories. One gram of grain provides at least 5 units of vitamins and 10 calories. If a gram of soybean meal costs 7 cents, a gram of meat byproducts 8 cents, and a gram of grain 9 cents, what mixture of these three ingredients will provide at least 54 units of vitamins and 60 calories per serving at minimum cost? What will be the minimum cost? The mixture should contain grams of soybean meal, grams of meat byproducts, and grams of grain, for a cost of cents. (Round your answer to one decimal place.)

Explanation / Answer

Given: Animal food must have

i.) At least 54 units of vitamins

ii.) At least 60 calories per serving

Here is how I setup the problem:

I Define these variables

s: number of grams of soybean meal

m: number of grams of meat-by-product

g: number of grams of grain

I use the information about vitamins to get this equation

2.5 s + 4.5 m + 5 g > = 54 [where "> =" means greater than or equal to]

I use the information about calories to get this equation

5s + 3m + 10g > = 60

I also create the price function (cost of the animal meal)

p = 8s + 9m + 10g [ This is the objective function which we are to minimize]


I use this constraints in the Linear Programming problem



2.5 s + 4.5 m + 5 g > = 54

5s + 3m + 10g > = 60

s > = 0, m > = 0, g >= 0


I solved the above equations as if they we the following equalities

2.5 s + 4.5 m + 5 g = 54

5s + 3m + 10g = 60

After solving the above system, I got

s = 0, m = 8 and g = 3.6

Plugging this into the cost function I get

p = 8(0) + 9(8) + 10(3.6)

= 108 cents

or $1.08

So I get the minimum cost is $1.08

The problem was looking for the mixture that would produce the minimum cost so from my
answer of s = 0, m = 8, and g = 3.6 the mixture would be

8 grams of meat by-products and 3.6 grams of grain.

Update: I solved this a couple of days ago with algebra, as shown above. Just now I solved the Linear Programming problem in Excel using Excel Solver and got the same results.

The mixture that minimizes the cost has

0 grams of soybean meal, 8 grams of meat-by-products, and 3.6 grams of grain.

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