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Math Help - Find Maximum Volume

  1. #1
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    Find Maximum Volume

    Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

    16 meters by 10 meters.

    Find the dimensions of the box that has the maximum volume.

    V=M^3

    Length = 10m
    Width = 16m
    Height = x


    Therefore the Function is:

    v=x(16-2x)(10-2x)
    v= x(160-12x-4x^2)
    v=(160x-12x^2-4x^3)

    Find the derivative and solve for v'=0.

    v'=160-24x-12x^2

    I'm stuck on solving v'=0.

    The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.


    I don't know what the next step is.
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  2. #2
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    Quote Originally Posted by Butum View Post
    Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

    16 meters by 10 meters.

    Find the dimensions of the box that has the maximum volume.

    V=M^3

    Length = 10m
    Width = 16m
    Height = x


    Therefore the Function is:

    v=x(16-2x)(10-2x)
    v= x(160-12x-4x^2)
    v=(160x-12x^2-4x^3)

    Find the derivative and solve for v'=0.

    v'=160-24x-12x^2

    I'm stuck on solving v'=0.

    The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.


    I don't know what the next step is.
    \sqrt{24^2-4(-12)(160)} > 0
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  3. #3
    MHF Contributor

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    Quote Originally Posted by Butum View Post
    Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

    16 meters by 10 meters.

    Find the dimensions of the box that has the maximum volume.

    V=M^3

    Length = 10m
    Width = 16m
    Height = x


    Therefore the Function is:

    v=x(16-2x)(10-2x)
    v= x(160-12x-4x^2)
    Your middle term is wrong. It should be -32x- 12x= -44x

    v=(160x-12x^2-4x^3)
    v= 160x- 44x^2- 4x^3

    Find the derivative and solve for v'=0.

    v'=160-24x-12x^2
    v'= 160- 88x- 12x^2

    I'm stuck on solving v'=0.

    The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.


    I don't know what the next step is.
    With the function corrected, the discriminant is 64. In fact, if you first divide by 4, the equation becomes reasonably easy to factor.
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