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Math Help - Linear Programming application

  1. #1
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    Linear Programming application

    My problem is with constraints.

    Here is the problem in a nutshell

    1000 guests in a fundraiser. $1 for juice $1.75 for Soda
    Constraints:

    each drink has to be served in an unused plastic cup of which there are 5,000

    juice must acount for at least 30% of drinks served

    each guest can only have 3 sodas

    each juice must have 2 icecubes and each soda must have 4 ice cubes, there are 15,000 icecubes available.

    goal is to maximize profit

    I know the objective function is going to be 1x+1.75y=z
    I think one constraint is going to be y less than/equal to 3 and after that i get lost. please help
    Last edited by Unusualtoe; September 20th 2007 at 02:05 PM.
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  2. #2
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    Hello, Unusualtoe!

    This certainly has a tricky set-up . . .


    1000 guests at a fundraiser.
    $1 for juice, $1.75 for soda
    Let x = number of servings of juice sold: . {\color{blue}x \geq 0}
    Let y = number of servings of soda sold: . {\color{blue}y \geq 0}



    Each drink has to be served in an unused plastic cup of which there are 5,000
    We have: . {\color{blue}x + y \:\leq \:5000}


    Juice must account for at least 30% of drinks served
    It says: .Juice is at least 30% of (Juice + Soda)

    So we have: . x \:\geq \:0.30(x+y)
    . . This becomes: . x \:\geq\:0.3x + 0.3y\quad\Rightarrow\quad {\color{blue}7x - 3y \:\geq\:0}



    Each guest can only have 3 sodas
    There are 1000 guests.
    If each has 3 sodas, 3000 sodas will be sold: . {\color{blue}y \:\leq\:3000}



    Each juice must have 2 ice cubes and each soda must have 4 ice cubes.
    There are 15,000 ice cubes available.
    There will be: . 2x + 4y ice cubes used.
    . . 2x + 4y \:\leq \:15,000\quad\Rightarrow\quad {\color{blue}x + 2y \:\leq \:7500}



    Goal is to maximize profit
    Objective function: . {\color{red}P \:=\:x + 1.75y}


    Graph the region determined by the six constrants (in blue).
    Test the vertices in the Profit function (in red).

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