document.write( "Question 1197925: What is the maximum number of bottles, each of diameter 8 cm, that can be packed into a box with a square base measuring 632 cm by 632 cm? (Using the hexagonal packaging pattern) \n" ); document.write( "

Algebra.Com's Answer #831387 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "What is the maximum number of bottles, each of diameter 8 cm, that can be packed
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document.write( "Usual \"square cells\" paking scheme gives the answer   =  = 6241 bottles.\r\n" );
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document.write( "\"Equilateral triangle cells\" packing scheme has the distance (the shift) between the lines of 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 lines of centers is\r\n" );
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document.write( "     =  =  = 90.06668399 (rounded to integer number 90).\r\n" );
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document.write( "It means that we have the sequence of numbers of bottles in paired rows as\r\n" );
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document.write( "    (78 + 79) + (78 + 79) + . . . + (78 + 79).      (1)\r\n" );
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document.write( "Each sum in parentheses in (1) is 78+79 = 157 bottles and the number of such groups of 157 bottles in (1) is 90/2 = 45.\r\n" );
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document.write( "Hence, the total number of bottles at such packing is  157*45 = 7065 bottles.    ANSWER\r\n" );
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document.write( "For completeness, compare the two numbers for two packing schemes.\r\n" );
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