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Math Help - "Find the volume of the largest open box.."

  1. #1
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    "Find the volume of the largest open box.."

    Here's the problem:

    "Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

    I have no provided diagram, the answer is 1024 cubic inches.

    Here's what I've done:

    V=x*y*z

    y = z, y*z = 24.

    y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.
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  2. #2
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    Quote Originally Posted by Wolvenmoon View Post
    Here's the problem:

    "Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

    I have no provided diagram, the answer is 1024 cubic inches.

    Here's what I've done:

    V=x*y*z

    y = z, y*z = 24.

    y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.
    let x = side length of squares cut from each corner

    H = x

    L = W = 24 - 2x

    V = LWH = (24-2x)^2 \cdot x

    V = (576 - 96x + 4x^2) \cdot x

    V = 576x - 96x^2 + 4x^3

    find \frac{dV}{dx} and maximize
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  3. #3
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    Quote Originally Posted by Wolvenmoon View Post
    Here's the problem:

    "Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

    I have no provided diagram, the answer is 1024 cubic inches.

    Here's what I've done:

    V=x*y*z

    y = z, y*z = 24.

    y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.
    Hi Wolvenmoon,

    as the same size square is being cut from all 4 corners,
    simply label the side of that removed square "x".

    Then, since 2 squares are being removed from any side,
    the base of the box is a square of sides 24-2x.
    The height of the box is x.

    The box volume of the bos is (base area)(height)=x(24-2x)(24-2x).

    You can either multiply this out and differentiate wrt x
    or use the product rule.

    \frac{dV}{dx}=0

    \frac{d}{dx}\left[x(24-2x)^2\right]=0

    2(24-2x)(-2)x+(24-2x)^2=0

    -4x(24-2x)+(24-2x)(24-2x)=0

    4x=24-2x

    6x=24\ \Rightarrow\ x=4

    V=4(24-8)^2=4(16^2)=4(256)=1024
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