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Math Help - Optimization problem.

  1. #1
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    Optimization problem.

    The length of a cedar chest is twice its width. The cost/dm of the lid is 4 times the cost/dm of the rest of the cedar chest. If the volume of the cedar chest is 1440 dm, find the dimensions so that the cost is a minimum.

    Unfortunately, I don't know where to start (didn't see any questions like this at all in my examples or the questions I've solved).
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  2. #2
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    Start by finding a function to represent the cost of making the chest...
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    Quote Originally Posted by Prove It View Post
    Start by finding a function to represent the cost of making the chest...
    How would I do that? That seems to be my only problem at the moment. Once I can do that, I can pretty much do the rest (differentiate, find where the derivative is 0, then plug back the minimum into the original equations to find the other unknowns). I know that LWH = 1440 dm, and that L = 2W. Plugging this back into the equation I get 2WH = 1440 and then I rearrange for H = \frac{720}{W^{2}} but that's about as far as I can get.
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    It should be pretty clear that the cost is the same as the surface area of an open box, plus 4 times that cost for the lid...
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    Quote Originally Posted by Prove It View Post
    It should be pretty clear that the cost is the same as the surface area of an open box, plus 4 times that cost for the lid...
    So, C = 2LW+2LH+2WH+4LW?
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    No, first evaluate the surface area for the OPEN box (i.e. without the lid...).

    Then the lid will be 4 times all of that...
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    Quote Originally Posted by Prove It View Post
    No, first evaluate the surface area for the OPEN box (i.e. without the lid...).

    Then the lid will be 4 times all of that...
    C = 4(2LH + 2WH + 2LW)?

    Thanks for all the help, by the way.
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  8. #8
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    No.

    Bottom: \displaystyle LW

    Sides: \displaystyle 2LH + 2WH

    4 times that: \displaystyle 4(LW + 2LH + 2WH).


    So total cost: \displaystyle LW + 2LH + 2WH + 4(LW + 2LH + 2WH).

    Now simplify and substitute your other formulas to make a function just of \displaystyle L or \displaystyle W.
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