document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1064392>Question 1064392</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a 6 inch diameter ball.  How many balls will fill a room 10ftx12ft with a 8ft ceiling (10x12x8=960)? This room has no furniture or obtrusion's to consider.<BR>\n" );
document.write( "I can visualize 8 balls occupying a 1ftx1ftx1ft space if the balls are set in straight rows from one another. 8 balls/cubic foot of space.<BR>\n" );
document.write( "8 balls x 960 = 7,680 balls<BR>\n" );
document.write( "BUT if I move the second level of balls over so they nestle into the depressed areas found between the first row of balls I am overwhelmed trying to figure that out.<BR>\n" );
document.write( "I have looked online and see there is rounded questimate of .74, but I am not sure if this is what I need?<BR>\n" );
document.write( "Do I simply take 8 balls/cubic ft  x 960 cubic ft divide by .74 = ?<BR>\n" );
document.write( "= 10,378 balls<BR>\n" );
document.write( "Thanks\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #679453 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=679453\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When packing spheres into a container,<BR>\n" );
document.write( "there is always \"wasted\" empty space between the spheres.<BR>\n" );
document.write( "There is also empty space between the spheres and the boundaries (floor, walls, ceiling) of the container.<BR>\n" );
document.write( "The volume of a sphere with 6 inch diameter (3 inch radius) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Volume%5Bball%5D=%284%2F3%29%2Api%2A%283in%29%5E3=36pi ALIGN=MIDDLE  ALT=\"Volume%5Bball%5D=%284%2F3%29%2Api%2A%283in%29%5E3=36pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3=about113.1 ALIGN=MIDDLE  ALT=\"in%5E3=about113.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3 ALIGN=MIDDLE  ALT=\"in%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A cube with 1 foot edges has a volume of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812in%29%5E3=1728 ALIGN=MIDDLE  ALT=\"%2812in%29%5E3=1728\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3 ALIGN=MIDDLE  ALT=\"in%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With the packing you visualized first,<BR>\n" );
document.write( "the fraction of the space filled with balls is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A36%2Api%2F1728=about0.5236=52.36%2F100=%2252.36%25%22 ALIGN=MIDDLE  ALT=\"8%2A36%2Api%2F1728=about0.5236=52.36%2F100=%2252.36%25%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You would fit only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A960=7680 ALIGN=MIDDLE  ALT=\"8%2A960=7680\"   DISABLED_style= \"max-width:90%; height:auto;\" > balls.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The alternate packing you propose is a better option,<BR>\n" );
document.write( "especially for filling a large room with relatively small balls.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C300%2C-4.2%2C5.8%2C-3%2C3%2C%0D%0Acircle%280%2C0%2C1%29%2Ccircle%282%2C0%2C1%29%2Ccircle%28-2%2C0%2C1%29%2C%0D%0Acircle%28-3%2C-1.732%2C1%29%2Ccircle%28-1%2C-1.732%2C1%29%2C%0D%0Acircle%281%2C-1.732%2C1%29%2Ccircle%283%2C-1.732%2C1%29%2C%0D%0Acircle%28-1%2C1.732%2C1%29%2Ccircle%281%2C1.732%2C1%29%2C%0D%0Agreen%28arrow%28-3.4%2C-1.7327%2C7%2C-1.732%29%29%2Cgreen%28arrow%28-2.4%2C0%2C7%2C0%29%29%2C%0D%0Aline%28-5%2C-2.732%2C6%2C-2.732%29%2Cred%28triangle%280%2C0%2C-1%2C-1.732%2C0%2C-1.732%29%29%2C%0D%0Ared%28triangle%280%2C0%2C-1%2C-1.732%2C1%2C-1.732%29%29%2Cred%28rectangle%280%2C-1.732%2C0.2%2C-1.532%29%29%2C%0D%0Aarrow%284.2%2C-1.732%2C4.2%2C0%29%2Carrow%284.2%2C0%2C4.2%2C-1.732%29%2C%0D%0Alocate%28-0.6%2C-1.73%2Cred%28R%29%29%2Clocate%280.4%2C-1.73%2Cred%28R%29%29%2C%0D%0Alocate%284.3%2C-0.7%2Csqrt%283%29R%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The height used up for one layer of balls is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29R=about1.732R ALIGN=MIDDLE  ALT=\"sqrt%283%29R=about1.732R\"   DISABLED_style= \"max-width:90%; height:auto;\" > rather than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2R ALIGN=MIDDLE  ALT=\"2R\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For a 1 cubic foot container, even with that strategy,<BR>\n" );
document.write( "you cannot achieve a very efficient packing of balls<BR>\n" );
document.write( "with a diameter of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6in=0.5ft ALIGN=MIDDLE  ALT=\"6in=0.5ft\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "  <BR>\n" );
document.write( "For the most efficient packing, the maximum theoretical fill ratio is,<BR>\n" );
document.write( "as you found, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.74=74%25 ALIGN=MIDDLE  ALT=\"0.74=74%25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can approach that ratio, if the size of your container<BR>\n" );
document.write( "is large compared to the volume of 1 sphere.<BR>\n" );
document.write( "The size of your container is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810ft%29%2A%2812ft%29%2A%288ft%29=960 ALIGN=MIDDLE  ALT=\"%2810ft%29%2A%2812ft%29%2A%288ft%29=960\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ft%5E3=960 ALIGN=MIDDLE  ALT=\"ft%5E3=960\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ft%5E3%2A%281728in%5E3%2Fft%5E3%29=1658880 ALIGN=MIDDLE  ALT=\"ft%5E3%2A%281728in%5E3%2Fft%5E3%29=1658880\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3 ALIGN=MIDDLE  ALT=\"in%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you could fill 74% of that space with balls, you would fill<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.74%2A1658880 ALIGN=MIDDLE  ALT=\"0.74%2A1658880\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3=1227571.2 ALIGN=MIDDLE  ALT=\"in%5E3=1227571.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in%5E3 ALIGN=MIDDLE  ALT=\"in%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The number of balls that fit in that \"fillable\" space is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1227571.2%2F113.1=10854.145 ALIGN=MIDDLE  ALT=\"1227571.2%2F113.1=10854.145\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2810854%29 ALIGN=MIDDLE  ALT=\"highlight%2810854%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > may be a good estimate.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For a more accurate calculation,<BR>\n" );
document.write( "accounting for additional space wasted along floor, ceiling and walls,<BR>\n" );
document.write( "we can calculates how many layers of balls fit into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8ft=96inches ALIGN=MIDDLE  ALT=\"8ft=96inches\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.1732%2A3in=5.196in ALIGN=MIDDLE  ALT=\"0.1732%2A3in=5.196in\"   DISABLED_style= \"max-width:90%; height:auto;\" > per layer.<BR>\n" );
document.write( "That would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=96%2F5.196=18.476 ALIGN=MIDDLE  ALT=\"96%2F5.196=18.476\"   DISABLED_style= \"max-width:90%; height:auto;\" > layers.<BR>\n" );
document.write( "That is 18 layers, with some wasted space above the top layer.<BR>\n" );
document.write( "Layers 1, 3, 5, etc would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2820+balls%29%2A%2824balls%29=480balls ALIGN=MIDDLE  ALT=\"%2820+balls%29%2A%2824balls%29=480balls\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Layers 2,,4,6,etc would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819balls%29%2A%2823balls%29=437balls ALIGN=MIDDLE  ALT=\"%2819balls%29%2A%2823balls%29=437balls\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "with 18 layers, you would have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2A%28480balls%29%2B9%2A%28437balls%29=highlight%288235balls%29 ALIGN=MIDDLE  ALT=\"9%2A%28480balls%29%2B9%2A%28437balls%29=highlight%288235balls%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is better than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7680balls ALIGN=MIDDLE  ALT=\"7680balls\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > layers of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=480balls ALIGN=MIDDLE  ALT=\"480balls\"   DISABLED_style= \"max-width:90%; height:auto;\" > per layer,<BR>\n" );
document.write( "but it is far less than the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10854balls ALIGN=MIDDLE  ALT=\"10854balls\"   DISABLED_style= \"max-width:90%; height:auto;\" > estimate </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );