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Math Help - Find volume of a cube as a function of diagonal length.

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    Find volume of a cube as a function of diagonal length.

    Find the relationship describing the volume (V) of a cube as a function of the length of the diagonal going through the cube (d). and evaluate it for a diagonal length of d = 1.2.
    Last edited by mr fantastic; September 29th 2010 at 02:01 PM. Reason: Re-titled.
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    Quote Originally Posted by choy777 View Post
    Find the relationship describing the volume (V) of a cube as a function of the length of the diagonal going through the cube (d). and evaluate it for a diagonal length of d = 1.2.
    Pythagoras' theorem will give you this.

    If you have a regular cube of side length "x", the volume is x^3

    The diagonal across the base square has length \sqrt{x^2+x^2}=\sqrt{2x^2}=\sqrt{2}x

    The diagonal going through the cube has length \sqrt{2x^2+x^2}=\sqrt{3x^2}=\sqrt{3}x

    \displaystyle\ V=x^3=\frac{\left(\sqrt{3}x\right)^3}{\left(\sqrt{  3}\right)^3}

    =\displaystyle\frac{d^3}{\left(\sqrt{3}\right)^3}=  \left(\frac{d}{\sqrt{3}}\right)^3

    You can then perform your calculation.
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    Lost

    I got lost how did it end up over root3 cubed
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    Quote Originally Posted by deadman View Post
    I got lost how did it end up over root3 cubed
    After applying Pythagoras' theorem twice, we get... "internal diagonal"= x\sqrt{3}

    where "x" is the length of a side of the cube.

    The volume of the cube is x^3

    Since \displaystyle\frac{\sqrt{3}}{\sqrt{3}}=1

    then \displaystyle\frac{(x\sqrt{3})^3}{(\sqrt{3})^3}=x^  3

    which is the cube volume.
    This allows us to express the cube volume as a function of the internal diagonal length.
    The internal diagonal goes from the bottom right-corner of the base of the cube to the
    top-left corner of the facing side,
    or from the bottom left-corner to the facing side's top right-corner.

    I'll be the first to admit that my sketch is not drawn well!
    The sides should appear to be the same lengths.
    Attached Thumbnails Attached Thumbnails Find volume of a cube as a function of diagonal length.-cube.jpg  
    Last edited by Archie Meade; September 30th 2010 at 11:50 AM.
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