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Math Help - how to find maximum volume of cube inside cone

  1. #1
    Gul
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    how to find maximum volume of cube inside cone

    I am finding this one tricky.

    Find the maximum volume of cube inscribed a right cone with radius 1, and height 3.

    Why can't I do 3/1 = 3-x/x

    If this was a cylinder, I'd have 3/1 = 3-h/r, and with a cube the sides are equal.
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  2. #2
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    Quote Originally Posted by Gul View Post
    I am finding this one tricky.

    Find the maximum volume of cube inscribed a right cone with radius 1, and height 3.

    Why can't I do 3/1 = 3-x/x

    If this was a cylinder, I'd have 3/1 = 3-h/r, and with a cube the sides are equal.
    Assuming you have defined x to be the sidelength of the cube then \frac{3}{1} = \frac{3 - x}{\frac{x}{2}} = \frac{2(3 - x)}{x}.
    Last edited by mr fantastic; June 13th 2009 at 06:15 PM. Reason: Fixed a typo.
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  3. #3
    Gul
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    why is it a 1/2 - x/2, and not just x/2?
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    Quote Originally Posted by Gul View Post
    why is it a 1/2 - x/2, and not just x/2?
    You're right. I mixed up my triangles and made a typo (too busy being funny to be careful). It's fixed.
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  5. #5
    Gul
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    They get this:

    3-X / 3 = Xsqrt(2) /2
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    Quote Originally Posted by Gul View Post
    They get this:

    3-X / 3 = Xsqrt(2) /2
    Oboy. It's my day for dumb mistakes today. I have no tme now but I'll show where that comes from later.

    Edit: The key is to note that when the cube is fit inside a cone, the longest side
    is the diagonal of its top face. Read:

    http://books.google.com.au/books?id=...esult&resnum=1

    http://www.math.hawaii.edu/home/putnam/1998.pdf
    Last edited by mr fantastic; June 13th 2009 at 07:31 PM.
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  7. #7
    Gul
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    Thanks!, just one question, why can you do the ratio I had before for a cylinder, but not for a cube
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  8. #8
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    Quote Originally Posted by Gul View Post
    Thanks!, just one question, why can you do the ratio I had before for a cylinder, but not for a cube
    The cross-section of a cylinder is a circle. The cross-section of a cube is a square. The distance from the centre of a circle to any point on the perimeter is always the same. This is not the case for a square.
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  9. #9
    Gul
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    Thank you. Are there any other rules which are important for inscribed solids?
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