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Math Help - Numbers of balls in a frustum

  1. #1
    williamlcs
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    Numbers of balls in a frustum

    I am unable to solve this question, helps needed !!

    I need to calculate how many balls can be put into the frustum.

    Basically i m using frustum of a cone formula to calculate the whole volume.

    Frustum Volume = (p * height * (RaČ + Ra * Rb + RbČ) / 3)
    = (p * 72 * (81+135+225) / 3 )
    = 33250.61665

    However for the oval shape item, i do not know how to calculate it. Any idea what formula i should use?

    My idea is using the (total frustum volume - oval shape volume) / 1.5 to get the numbers of balls that can be place in the glass. Is my assumption right? I was told the number of balls can be place in is within the range of 3000 to 8000. Please advice.
    Attached Thumbnails Attached Thumbnails Numbers of balls in a frustum-img_52971-copy.jpg  
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  2. #2
    Senior Member
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    Quote Originally Posted by williamlcs View Post
    I am unable to solve this question, helps needed !!

    I need to calculate how many balls can be put into the frustum.

    Basically i m using frustum of a cone formula to calculate the whole volume.

    Frustum Volume = (p * height * (RaČ + Ra * Rb + RbČ) / 3)
    = (p * 72 * (81+135+225) / 3 )
    = 33250.61665

    However for the oval shape item, i do not know how to calculate it. Any idea what formula i should use?

    My idea is using the (total frustum volume - oval shape volume) / 1.5 to get the numbers of balls that can be place in the glass. Is my assumption right? I was told the number of balls can be place in is within the range of 3000 to 8000. Please advice.
    This is a sphere-packing problem. A Google search turns up several useful links (second and third) which you should read to get an idea for what's involved in the problem. Your assumption for the formula is not correct.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by williamlcs View Post
    I am unable to solve this question, helps needed !!

    I need to calculate how many balls can be put into the frustum.

    Basically i m using frustum of a cone formula to calculate the whole volume.

    Frustum Volume = (p * height * (Ra² + Ra * Rb + Rb&#178 / 3)
    = (p * 72 * (81+135+225) / 3 )
    = 33250.61665

    However for the oval shape item, i do not know how to calculate it. Any idea what formula i should use?

    My idea is using the (total frustum volume - oval shape volume) / 1.5 to get the numbers of balls that can be place in the glass. Is my assumption right? I was told the number of balls can be place in is within the range of 3000 to 8000. Please advice.
    It seems to me that you don't have (or haven't given us) sufficient
    information to calculate the volume of the glass.

    Are the dimples parts of the surface of spheres? How many are there?
    How far are their centres from the axis of the glass? how far below the
    rim are the centres of the spheres?

    Even with this information I would not want to calculate the volume
    of this glass!

    (Having said that it seems that 3-D hit-or-miss Monte-Carlo integration
    might not be too bad for this - actually it would be quite neat)

    After having calculated the volume you will still need to use the packing
    fraction you will find from following JakeD's link to find the number of
    balls.

    RonL
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