word problem: company gets milk from two dairies and blends

sallyk57

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Each week, the DeeLite Milk Company gets milk from two dairies and blends the milk to get the desired amount of butterfat for the company's premier product. Dairy A can supply at the most 700 tons of milk averaging 3.7% butterfat and costing $240 per ton. Dairy B can supply milk averaging 3.2% butterfat and costing $200 per ton. How much milk from each supplier should DeeLite use to get 1000 tons of milk with at least 3.5% butterfat at minimum cost? What is the total cost?

I don't know where to start.
 
Re: word problem: company gets milk from two dairies and ble

sallyk57 said:
I don't know where to start.
Has your class not covered "linear programming" yet...?

sallyk57 said:
Each week, the DeeLite Milk Company gets milk from two dairies and blends the milk to get the desired amount of butterfat for the company's premier product. Dairy A can supply at the most 700 tons of milk averaging 3.7% butterfat and costing $240 per ton. Dairy B can supply milk averaging 3.2% butterfat and costing $200 per ton. How much milk from each supplier should DeeLite use to get 1000 tons of milk with at least 3.5% butterfat at minimum cost? What is the total cost?
The following won't make much sense until you've learned about solving and graphing linear equations and inequalities, and then learned about linear programming. But this is the usual solution process:

Pick variables for the amounts (in tons) of milk bought from each dairy; say, "A" and "B".

Write down the basic physical constraints for each variable. (Hint: Can you buy negative tonnage?)

Note that A < 700, and that A + B must equal 1000. Then B > 300.

Write expressions for the milk-fat in each purchase. (Hint: If the milk were five percent butterfat, then "x" tons would contain 0.05x tons of fat.) Sum these "fat" expressions, and set "greater than or equal to" the minimum required fat content.

Write expressions for the cost of each purchase. (Hint: If the milk were $150 per ton, then "x" tons would cost 150x dollars.) Sum these "cost" expressions; this is your optimization equation.

Graph the constraints. Solve for the coordinates of the feasibility region. Plug them into your optimization equation. Whichever point gives you the smallest result is the answer.

Eliz.
 
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