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Math Help - Linear Programming Help Please?

  1. #1
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    Post Linear Programming Help Please?

    I need help trying to solve to problems, I want to know the answer to them first and then I want to know how to solve them.

    I want to know if anyone can post a step by step how to solve at least one of these in great detail, or if there is a site that can explain these in great detail. I am very bad at math and last year I had the same questions on Algebra 2, and I completely forgot how to do them this year.

    1) A tray of corn muffins requires 4c milk and 3c wheat flour. A tray of bran muffins takes 2c milk and 3c wheat flour. There are 16c milk and 15 wheat flour available, the baker makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each should he make in order to maximize profits?




    2) Each gallon of garlic dressing requires 2qt oil and 2qt vinegar. Each gallon of tofu dressing requires 3qt oil and 1qt vinegar. Jim makes $3 profit on each gallon of garlic dressing and $2 profit on each gallon of tofu dressing. He has 18qt oil and 10qt vinegar on hand. How many gallons of each type of dressing should he make to maximize profits?




    I need step by step explanation on at least one, so I can remember how to solve these types of problems.
    Last edited by Brazuca; September 8th 2007 at 07:51 PM. Reason: Highlight the numbers.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Brazuca View Post

    1) A tray of corn muffins requires 4c milk and 3c wheat flour. A tray of bran muffins takes 2c milk and 3c wheat flour. There are 16c milk and 15 wheat flour available, the baker makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each should he make in order to maximize profits?
    First introduce variables C >=0 and B>=0 for the number of trays of corn
    and bran muffins produced.

    Now the amount of milk used for this production is:

    C*0.04+B*0.02 <= 0.16

    this is less than or equal to 0.16 as 16c is the total milk available.

    Now we do the same for wheat flour:

    C*0.03+B*0.03 <= 0.15

    Thus we have the constraints on production:

    C >= 0
    B >= 0
    C*0.04+B*0.02 <= 0.16
    C*0.03+B*0.03 <= 0.15

    We want to maximise the profit 3C + 2B.

    Now I am sure that you have been shown a graphical method of solving this,
    or been told that the optimum occurs at a vertex of the feasible region
    defined by the constraints.

    RonL
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  3. #3
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    Hello, Brazuca!

    2) Each gallon of garlic dressing requires 2qt oil and 2qt vinegar.
    Each gallon of tofu dressing requires 3qt oil and 1qt vinegar.
    Jim makes $3 profit on each gallon of garlic dressing
    . . and $2 profit on each gallon of tofu dressing.
    He has 18qt oil and 10qt vinegar on hand.
    How many gallons of each type of dressing should he make to maximize profit?

    Tabulate the given information.

    \begin{array}{cccccccc} & | & \text{oil} & | & \text{vinegar} & | & \text{profit} & | \\ \hline<br />
\text{Garlic} & | & 2 & | & 2 & | & \$3 & | \\<br />
\text{Tofu} & | & 3 & | & 1 & | & \$2 & | \\ \hline<br />
& & 18 & & 10 & <br />
\end{array}

    Let x = number of gallons of Garlic dressing: .  x \geq 0\;\;[1]
    Let y = number of gallons of Tofu dressing: . y \geq 0\;\;[2]

    \begin{array}{ccccc}\text{Oil used:} & 2x+ 3y & \leq & 18 & [3]\\<br />
\text{Vinegar used:} & 2x + y & \leq & 10 & [4]\end{array}


    From [1] and [2], the region is in Quadrant 1.

    Graph the line of [3]: . 2x + 3y \:=\:18
    It has intercepts (9,0) and (0,6)
    Draw the line and shade the region below the line.

    Graph the line of[4]: . 2x + y \:=\:10
    It has intercepts (5,0) and (0,10)
    Draw the line and shade the region below the line.


    The shaded region looks like this:
    Code:
            |
         10 *
            | *
            |   *
          6 o     *
            |::::o  *
            |:::::::::o
            |:::::::::::*  o
          - + - - - - - - * - - o - -
            |             5     9

    The critical points are the vertices of this region.

    We know three of them: . (0,0),\:(5,0),\:(0,6)

    The fourth is the intersection of the lines of [3] and [4].
    Solve the system and we get: . (3,4)


    Test these vertices in the profit function, P \:=\:3x + 2y
    . . to determine which produces maximum profit.

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  4. #4
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    Talking Thanks.

    Thanks, alot guys that really helped me.

    *Currently using this page to help me post math equations using the MATH MATH*

    Green Y = e^{-x}
    Red  Y - e^{2x}
    Blue X = 1

    So Integral of (e^{2x} - e^{-x}

    ((e^{2}/2) + (e^{-1})) -((e^{0}/2) + (e^{-0}))

    (e^{2}/2) + (e^{-1}) -(1/2) + (1)

    (e^{2}/2) + (e^{-1}) -(1/2) + (1)
    Last edited by Brazuca; September 7th 2009 at 12:39 PM.
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