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Math Help - Maximize Dimensions of an open box

  1. #1
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    Maximize Dimensions of an open box

    A box with a square base and open top must have a volume of 442368 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

    First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base.
    [Hint: use the volume formula to express the height of the box in terms of x.]
    Simplify your formula as much as possible.

    can anyone tell me what this equation is? );
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tyuolio View Post
    A box with a square base and open top must have a volume of 442368 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

    First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base.
    [Hint: use the volume formula to express the height of the box in terms of x.]
    Simplify your formula as much as possible.

    can anyone tell me what this equation is? );
    This is using calculus

    Ok, so it is always a good idea to draw a diagram when we can. see the diagram below.

    let the length of each side of the base of the box be x
    let the height of the box be y

    then the volume is given by:

    V = length*width*height = x^2 * y
    we want the volume to be 442368

    so we have x^2 * y = 442368
    we want everything in terms of x, so let's solve for y
    => y = 442368/(x^2)

    Now the surface area is the area of all the faces (the top not included since it's an open top). so the surface area is given by:

    S = x^2 + xy + xy + xy + xy = x^2 + 4xy
    => S = x^2 + 4x(442368/(x^2))
    => S = x^2 + 1769472/x

    we want S to be a minimum.
    for minimum we find S' and set it to zero:

    S' = 2x -1769472x^-2

    for min, set S' = 0
    => 2x - 1769472x^-2 = 0
    => 2x^3 - 1769472 = 0
    => 2x^3 = 1769472
    => x^3 = 1769472/2 = 884736
    => x = cuberoot(884736)
    => x = 96

    so the base of the box must be 96 for minimum surface area. what must the height be?

    y = 442368/(x^2) = 442368/(96^2) = 48

    so the dimensions of the box for minimum surface area are:
    width = length = 96 and height = 48
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  3. #3
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    thank you so much
    yes, i'm in calculus. that was what i needed.
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