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Math Help - Hard Math Question!

  1. #1
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    Hard Math Question!

    The owner os a small Nutt's & Nuts company makes and sells two types of 5-ounce mixed nuts packages. A low-grade mixture containing 3 ounces of cashews and 2 ounces of peanuts and a high-grade mixture containing 4 ounces of cashews and 1 ounce of peanuts. The profit is $.70 on each package of the low-grade mixture and $.80 on each package of the high-grade mixture. The mixing machine that gets filled every hour can hold utmost 36 ounces of cashews and no more than 14 ounces of peanuts. By graphing, find: a) the maximum hourly profit, and b) the production schedule that maximizes the hourly profit.

    Please help with graph points and answers!!! Thanks!
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  2. #2
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    Hello, Greenbaumenom!

    Have you never done a Linear Programming problem?


    A company makes and sells two types of 5-ounce mixed nuts packages.
    A low-grade mixture containing 3 ounces of cashews and 2 ounces of peanuts
    and a high-grade mixture containing 4 ounces of cashews and 1 ounce of peanuts.

    The profit is $0.70 on each package of the low-grade mixture
    and $0.80 on each package of the high-grade mixture.

    The mixing machine that gets filled every hour can hold at most 36 ounces of cashews
    and no more than 14 ounces of peanuts.

    By graphing, find:
    a) the maximum hourly profit
    b) the production schedule that maximizes the hourly profit.

    Why do they always ask these questions backwards ??
    Organize the data . . .

    . . \begin{array}{c|c|c|}<br />
& \text{cashews} & \text{peanuts} \\ \hline<br />
\text{Low }(x) & 3 & 2 \\<br />
\text{High }(y) & 4 & 1 \\ \hline<br />
\text{Total} & 36 & 14 \end{array}


    We have: . \begin{array}{cc} x \:\geq\: 0 & {\color{blue}[1]}\\ y \: \geq\: 0 & {\color{blue}[2]}\\<br />
3x + 4y \:\leq\: 36 & {\color{blue}[3]}\\ 2x + y \:\leq\: 14 & {\color{blue}[4]}\end{array}

    Profit Function: . P \:=\;0.70x + 0.80y



    [1] and [2] places us in Quadrant 1.

    The line of [3] is: . 3x+4y \:=\:36
    . . It has intercepts (12,0) and (0,9).
    . . Graph the line and shade the region below the line.

    The line of [4] is: . 2x + y \:=\:14
    . . It has intercepts (7,0) and (0,14).
    . . Graph the line and shade the region below the line.

    The graph looks like this:
    Code:
            |
         14 *
            |*
            | *
            |  *
          9 o   *
            |::* *
            |:::::o
            |::::::* *
            |:::::::*   *
            |::::::::*     *
        - - o - - - - o - - - * - -
            |         7      12

    The vertices of the shaded region are: . (0,0),\;(7,0),\;(0,14)
    . . and the intersection of the two lines: (4,6)

    Test them in the Profit Function to see which produces maximum profit.

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