document.write( "Question 1147250: What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 792 cm by 792 cm? \n" ); document.write( "

Algebra.Com's Answer #768572 by ikleyn(52800) $\"\"$ $\"About$

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document.write( "Usual \"square cells\" paking scheme gives the answer   =  = 9801 bottles.\r\n" );
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document.write( "\"Equilateral triangle cells\" packing scheme has the distance (the shift) between the centers of every two adjacent rows of bottles  \r\n" );
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document.write( "    h =  = 6.9282 cm;\r\n" );
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document.write( "Hence, the number of such shifts is\r\n" );
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document.write( "     =  =  = 113.1607 (rounded to integer number 113).\r\n" );
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document.write( "It means that we have the sequence of numbers of bottles in rows as\r\n" );
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document.write( "    99 + 98 + 99 + 98 + . . . + 98.      (1)\r\n" );
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document.write( "Notice that each sign \" + \" corresponds to one shift between the rows ---- hence we have 114 addends in the sum (1).\r\n" );
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document.write( "Then we can group them in 114/2 = 57 pairs of rows with (99+98) = 197 bottles in each pair of rows.\r\n" );
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document.write( "Hence, the total number of bottles at such packing is  57*197 = 11229 bottles.\r\n" );
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document.write( "Now compare the two numbers for two packing schemes and make your decision.\r\n" );
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\n" ); document.write( "comment from student: Hello, about question 1147250. What is the maximum number of bottles, each of diameter 8 cm,
\n" ); document.write( "that can be packed into a box with a square base measuring 792 cm by 792 cm? My teacher told me that
\n" ); document.write( "the tightest possible method is a hexagonal shape like so.
\n" ); document.write( "https://imgur.com/a/UIt6c5o Could you please solve it using this method of packing thx.
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\n" ); document.write( "\n" ); document.write( "My response : Good question; thanks for asking.\r
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\n" ); document.write( "\n" ); document.write( "My second packing scheme \"Equilateral triangle cells\" is exactly that tightest possible hexagonal shape
\n" ); document.write( "mentioned by your teacher.\r
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\n" ); document.write( "\n" ); document.write( "It has 99 bottles in the longest row and 98 bottles in the next adjacent row.\r
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\n" ); document.write( "\n" ); document.write( "So, you just have this solution (!) \r
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\n" ); document.write( "\n" ); document.write( "Make a sketch to better understand it (!)\r
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\n" ); document.write( "\n" ); document.write( "Come again soon to the forum to learn something new (!)\r
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