an animal food must provide at least 57 units of vitamins and 63 calories per se

Question

an animal food must provide at least 57 units of vitamins and 63 calories per serving. each serving should contain at least 15 grams. one gram of soybean meal, meat byproducts, and grain provide respectively 2.5,5, and 4 units of vitamins respectively. one gram of soybean meal, meat byproducts, and grain has 4 calories, 3 calories, and 8 calories, respectively. a gram of soybean meal costs 9cents, a gram of meat byproducts costs 11 cents, and a gram of grain costs 8 cents. what mixture of these ingredients will provide the needed vitamins and calories for the minimum cost?

I just need the equations. Thank you for the help.

Explanation / Answer

At least 57 units of vitamins and 63 calories per serving.

each serving should contain at least 15 grams.

1 g of soybean has 2.5 units of vitamins and 4 cals and costs 9 cents
1 g of meat has 5 units of vitamins and 3 cals and costs 11 cents
1 g of grain has 4 units of vitamins and 8 cals costs 8 cents

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

1 g of soybean costs 9 cents
So, x grams costs 9x cents

And y grams of meat costs 11y cents

And z grams of grain costs 8z cents

Total cost = C say

C = 9x + 11y + 8z cents -----> to be minimized

Atleast 57 units of vitamins :
1 g of soybean has 2.5 units of vitamins
So, x g of soybean has 2.5x units of vitamins

And y g of meat has 5y units of vitamins

And z g of grain has 4z units of vitamins

So, 2.5x + 5y + 4z >= 57 ----> FIRST EQUATION

Atleast 63 cals :
1 g of soybean has 4 cals
So, x g has 4x cals

Similarly, y g of meat ---> 3y cals

And z g of grain ---> 8z cals

And thus, 4x + 3y + 8z >= 63 ---> SECOND EQUATION

Clearly, x,y and z are grams, so they have to be non-negative

So, x >= 0 , y >= 0 and z >= 0

So, here is the final answer :

Objective function to be minimized : C = 9x + 11y + 8z cents -----> to be minimized

Constraints :
2.5x + 5y + 4z >= 57 ---> vitamin equation
4x + 3y + 8z >= 63 ---> calorie equation
x >= 0
y >= 0
z >= 0

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