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Math Help - Methods and Matrices.

  1. #1
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    Methods and Matrices.

    So here's the problem:
    A small pastry company is trying to determine the optimal number of products to make in a given week. The company's three most popular products and Chocolate Muffins, Eclairs, and Blueberry Scones. Each product requires preparation, baking, and packaging time. Each of the previously mentioned departments operates at a maximum of 380, 330, and 120 labor-hours per week respectively.

    Chocolate Muffins Eclairs Blue Berry Scones
    Rreparation 0.5hr 1.0hr 1.5hr
    Baking 0.6hr 0.9hr 1.2hr
    Packaging 0.2hr 0.3hr 0.5hr

    A) Determine a system of equations, represent in matrix form, that would maximize company production.

    [.5 .6 .2 380]
    [1.0 .9 .3 330]
    [1.5 1.2 .5 120]

    B) Reduce the matrix from A to determine the optimal number of each product to be produced in a given week.

    [1 0 0 -960 ]
    [0 1 0 1966.66]
    [0 0 1 -1600 ]



    So I clearly went wrong somewhere but I'm not sure where or what to do.
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  2. #2
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    Re: Methods and Matrices.

    Quote Originally Posted by TimK View Post
    So here's the problem:
    A small pastry company is trying to determine the optimal number of products to make in a given week. The company's three most popular products and Chocolate Muffins, Eclairs, and Blueberry Scones. Each product requires preparation, baking, and packaging time. Each of the previously mentioned departments operates at a maximum of 380, 330, and 120 labor-hours per week respectively.

    Chocolate Muffins Eclairs Blue Berry Scones
    Rreparation 0.5hr 1.0hr 1.5hr
    Baking 0.6hr 0.9hr 1.2hr
    Packaging 0.2hr 0.3hr 0.5hr

    A) Determine a system of equations, represent in matrix form, that would maximize company production.

    [.5 .6 .2 380]
    [1.0 .9 .3 330]
    [1.5 1.2 .5 120]

    B) Reduce the matrix from A to determine the optimal number of each product to be produced in a given week.

    [1 0 0 -960 ]
    [0 1 0 1966.66]
    [0 0 1 -1600 ]



    So I clearly went wrong somewhere but I'm not sure where or what to do.
    For starters, you haven't written matrix equations. You should actually have written

    \displaystyle \begin{align*} \left[\begin{matrix} 0.5 & 1.0 & 1.5 \\ 0.6 & 0.9 & 1.2 \\ 0.2 & 0.3 & 0.5  \end{matrix}\right]\left[\begin{matrix} m \\ e \\ s \end{matrix}\right] = \left[\begin{matrix} 380 \\ 330 \\ 120 \end{matrix}\right] \end{align*}
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