One gram of soybean meal provides at least 2.5 units of vitamins and 5 calories.

Question

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 post at least 5 units of vitamins and 10 calories. If a gram of soybean meal costs 6 cents, a gram of meat byproducts 8 cents, and a gram of pain 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 mine should contain [ ] grams of soybean meal, [ ] grams of meat byproducts, and [ ] grams of pain for a cost of [ ]cents Round your answer et to one decimal place )

Explanation / Answer

Soybean : 2.5V and 5C
Soybean cost = 6 cents
Meat : 4.5V and 3C
Meat cost = 8 cents
Grain : 5V and 10C
Grain cost = 9 cents

We need 54V and 60C

Say x grams of soybean , y grams of meat and z grams of grain

2.5x + 4.5y + 5z = 54 --> ONE
5x + 3y + 10z = 60 --> TWO

Equation ONE multiplied by 2 :

5x + 9y + 10z = 108
5x + 3y + 10z = 60

Subtracting :

6y = 48

y = 8

So, equation TWO becomes :

5x + 3y + 10z = 60
5x + 24 + 10z = 60
5x + 10z = 36

Now, here clearly, soybean costs the least, but with soybean alone, we'd need 36/5 = 7.2 grams, which would cost 7.2 * 6 = 43.2 cents

With grain alone, we'd need 36/10 = 3.6 grams, which would cost 3.6 * 9 = 32.4 cents, clearly the cheaper option

Cost = 6x + 8y + 9z --> to be minimized

So, x = 0 , y = 8 and z = 3.6

Cost = 6(0) + 8(8) + 9(3.6)
Cost = 0 + 64 + 32.4
Cost = 96.4

So, here is what you enter :

0 grams of soybean
8 grams of meat byproducts
3.6 grams of grain
cost of 96.4 cents

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