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Math Help - number of gumballs

  1. #1
    Member Veronica1999's Avatar
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    number of gumballs

    A spherical globe, 12 inches in diameter, is filled with spherical gumballs, each having a one inch diameter. Estimate the number of gumballs in the globe, and explain your reasoning.

    The volume of the globe is 904.32 and the volume of each gumball is 0.523333.
    This allows 1728 gumballs to fit in the globe.
    This doesn't seem right.
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  2. #2
    Super Member TheChaz's Avatar
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    Re: number of gumballs

    There is "dead space" (i.e. just air) in between gumballs. If they were somhow liquified, then we could just divide the total volume (~900) by the volume per gumball (~.5) to get ~1800.

    One approach would be to take a rough average between our first guess (1800, or 1728...) and the following:
    Imagine 1 (in^3) cubes. We could fit ~900 in there, so the average would be around 2350.
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  3. #3
    Grand Panjandrum
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    Re: number of gumballs

    Quote Originally Posted by Veronica1999 View Post
    A spherical globe, 12 inches in diameter, is filled with spherical gumballs, each having a one inch diameter. Estimate the number of gumballs in the globe, and explain your reasoning.

    The volume of the globe is 904.32 and the volume of each gumball is 0.523333.
    This allows 1728 gumballs to fit in the globe.
    This doesn't seem right.
    Kepler problem/conjecture/theorem?

    Its consequence is that the maximum packing density of spheres is ~74%.

    In this case you will not be able to obtain this because of the finite volume of the container and the irregularity of the packing. Experimental results for random close packing give a maximum of ~50% and theory says that random close packing cannot exceed about 64%.


    CB
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