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Thread: Maximization using Lagrange multipliers, word problem

  1. #1
    Super Member maxpancho's Avatar
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    Maximization using Lagrange multipliers, word problem

    A manufacturer makes two models of an item, standard and deluxe. It costs \$40 to manufacture the standard model and \$60 for the deluxe. A market research firm estimates that if the standard model is priced at $x$ dollars and the deluxe at $y$ dollars, then the manufacturer will sell $500(y−x)$ of the standard items and $45000+500(x−2y)$ of the deluxe each year. How should the items be priced to maximize profit?

    Seems like they want to solve this using Lagrange multipliers (is it even possible?)

    $P(x,y)=500(y-x)(x-40)+(45000+500(x-2y))(y-60)$

    What could be the constraint?
    Last edited by maxpancho; Feb 15th 2015 at 05:28 AM.
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    Re: Maximization using Lagrange multipliers, word problem

    You can solve this without using Lagrange multipliers. Just set the Gradient of the profit function to 0 and solve for x and y. Then just check to make sure you found a maximum profit not a minimum one.
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    Super Member maxpancho's Avatar
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    Re: Maximization using Lagrange multipliers, word problem

    But how would I do that with Lagrange multipliers? It seems like it's easy to solve without them, but it was in the chapter about Lagrange multipliers. So I don't know...
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    Re: Maximization using Lagrange multipliers, word problem

    You could choose the constraints that you must sell a non-negative number of items. So $x$ and $y$ must satisfy that each of those market estimate numbers are non-negative.
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