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Math

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The surface area and the volume of a sphere are both 3 digit integers times π. If r is the radius of the sphere, how many integral values can be found for r?

  • Math - ,

    4 pi r^2 = (xyz)pi
    (4/3) pi r^3 = (ijk) pi

    4 r^2 = xyz
    (4/3) r^3 = ijk

    (4/3) r^3 is an integer less than 1000
    and greater than 99

    if (4/3) r^3 = 1000 r is 9.08 so try 9
    if r = 9, (4/3)r^3 = 972 , the biggest we can have for r^3 criterion

    now if (4/3) r^3 = 100 r is 4.21
    so smallest r for this criterion is
    r = 5
    so far we have
    5 , 6, 7, 8 , 9
    BUT we need to do the area thing
    4 r^2 = xyz
    if xyz = 1000, r = 15.8
    So
    15 is the biggest for the area
    if xyz = 100, r = 5 exactly
    so between 5 and 15
    so
    5,6,7,8,9 seem to work, five of them

  • Math - ,

    But 4/3 r^3 must be an integer, so r must be a multiple of 3.

    I'd say only 6 and 9 are available

  • Math - ,

    Yes, of course, sorry

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