Anonymous asked in Science & MathematicsMathematics · 8 years ago

Help with some linear math problem!?

Ok this is a math project I have to turn in. So if anyone has some input I'd really appreciate it!

A bottler uses pineapple, orange and grapefruit juice to make up two juice mixtures, orange-pineapple and orange-grapefruit. The mixtures are sold in quart bottles and the bottles make a profit of $0.50 per bottle of orange pineapple and $0.40 per bottle of orange-grapefruit. Each juice mixture is made by mixing equal amounts of the two juices in its name. Today there are 250 gallons of orange juice, 175 gallons of pineapple juice and 100 gallons of grapefruit juice available. (1 gallon = 4 quarts) Formulate and solve mathematically the problem of determining how many quart bottles of each juice mixture he bottler should produce to maximize profit.

So far I have:

let x= # of quarts of orange pineaplle

let y = # of quarts of orange grapefruit

To maximize P: P=$0.50x+$0.40

I have to create a product resource chart and the list of contraints. Then graph the constraints.

1 Answer

  • 8 years ago
    Favorite Answer

    You're on the right track so far. You need your constraints. These are:

    You can't produce negative quantities of juice, so:

    x >= 0

    y >= 0

    You only have 175 gallons = 700 quarts of pineapple juice, and since every quart of orange-pineapple juice is half pineapple, you can only make 1400 quarts of orange-pineapple:

    x <= 1400

    You only have 100 gallons = 400 quarts of grapefruit juice, and since every quart of orange-GF juice is half GF juice, so you can only make 800 quarts of orange-gf:

    y <= 800

    You only have 250 gallons = 1000 quarts of OJ, and every gallon of each juice is half OJ, you can only make 2000 quarts of the combined numbers of each juice:

    x + y <= 2000

    Now, if you graph this out, you see that you have a polytope (region of space made up of straight lines) with 5 points (corners where the boundary lines meet). The maximum is going to lie at one of those corners, so all you have to do is find the values of x and y at each of the corners, and then substitute those values directly into the formula for P and see which one gives you the maximum, and that's the answer to the problem. You actually, in this case, only have to check four corners, because one of the corners will be (0,0), and obviously at that point you make $0, so that's definitely not going to be the maximum. (Indeed, you actually only have to check two points - x=1400 and y = 600, and x=1200 and y=800. See if you can rationalize why by looking at the graph.)

    Source(s): PhD math student who did his Master's thesis in integer linear programming and optimization
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