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Math Help - cost minimization

  1. #1
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    cost minimization

    I need some help with a problem involving the optimization of area.

    This is the prompt to the problem:


    "A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides will be made of material costing $1 per square foot, and the top is to be constructed of glass costing $5 per square foot. What are the dimensions of the box of greatest volume than can be constructed for $72?"


    I have worked several optimization problems both involving cost and not but in this problem I am given the cost and not the area. I understand the process of optimization: Construct an equation that models the problem (hopefully with a sketch), use the given value in its respective equation and solve for one of the variables, plug this quantity into the main equation and solve for the other variable.

    I guess I should start with checking if my equation is correct.

    Surface Area = sides + top + bottom

    SA = 4xy + x^2 + x^2 Volume = x^2*y

    Cost = 4xy(1) + x^2(5) + x^2(1) Given value : $72 (max cost)

    Cost = 4xy + 5x^2 + x^2


    With problems like this I usually just take the derivative and doing that gave me the wrong dimensions. So going back I think that I need to set the cost equation equal to 72. However that would then incorporate implicit differentiation into an optimization problem and I have never done that before, just implicit by itself. Am I headed in the right direction and if so then what?
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  2. #2
    Member Greengoblin's Avatar
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    Let x be depth, y be width, and z be height. Then,

    xy+2xz+2yz+5xy=72

    2xz+2yz+6xy=72

    You can solve this using partial derivatives, by writing volume as a function of only two variables x, and y, and taking \nabla f =0.

    Or you can use the AM-GM inequality:

     \frac{2xz+2yz+6xy}{3}\ge(2xz\cdot 2yz\cdot 6xy)^{\frac{1}{3}}

    \frac{72}{3}\ge(2xz\cdot 2yz\cdot 6xy)^{\frac{1}{3}}

    24^3\ge 2xz\cdot 2yz\cdot 6xy

    24^3\ge 24x^2y^2z^2

    24^2\ge x^2y^2z^2

    24\ge xyz

    V \le 24 \implies V_{max}=24\text{ft}^3

    (The dimensions of your box are the set \{x,y,z\in\mathbb{R}:xyz=24\} )
    Last edited by Greengoblin; December 14th 2008 at 04:10 PM.
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  3. #3
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    Thank you for your help Greengoblin. Your final answer is correct according to my answer sheet but you created one more variable than is needed Greengoblin. I always learned that when a square base is involved one variable should be used for both of them. that is why my equation for volume is V=x^2*y. Also I have not learned about the AM-GM inequality so that is probably not a method I need to know yet. Also I'm not sure as to how I would solve it with partial derivatives.
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  4. #4
    Member Greengoblin's Avatar
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    You're welcome. I mis-read the part about having a square base as having a rectangular base, so replacing any instances of xy in the above calculation with x^2, should give more appropriate expressions as you pointed out. (This is also why I though partial derivatives would be needed..If you haven't heard of them yet, don't worry..their concerned with differentiating functions of several variables, which would have been the case had we had three independent dimensions.)

    The calculus way is:

    x^2+4xy+5x^2=72

    6x^2+4xy=72

    Now you can solve for y,

    y=\frac{72-6x^2}{4x}

    ...and get volume as a function of x only, by substitution:

    V(x,y)=x^2y\implies V(x)=\frac{72x^2-6x^4}{4x}=18x-\frac{3}{2}x^3

    Now you can find f'(x)...
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  5. #5
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    you know I just realized why I wasn't getting the problem right. I was taking the right approach but seeing the way you set up the equation allowed me to see where my mistake was.

    I must have confused the surface area equation with the cost equation after incorporating the $1 and $5 values for their respective faces. I took the derivative of this equation thinking the top and bottom quantities need to stay separate:

    [IMG]file:///C:/Users/Robert/AppData/Local/Temp/moz-screenshot-2.jpg[/IMG]

    Instead of adding them like you did and taking the derivative of this equation:



    That's what threw me off all along. I considered doing that earlier apparently didn't follow through. Wow definitely one of those "oh......duh" moments. Thank you again I appreciate the help.
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  6. #6
    Member Greengoblin's Avatar
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    Ok, glad you got it
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