Algebra.Com's Answer #847125 by ikleyn(52778)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "What is the maximum number of bottles, each of diameter 9 cm, that can be packed into a box
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document.write( "with a square base measuring 990 cm by 990 cm?
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document.write( "The diagram below shows the closest packing method.
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document.write( "https://ibb.co/x1WDp3H
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document.write( "Let's consider this dense packing scheme, which is shown in the attached diagram.\r\n" );
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document.write( "First horizontal row of bottles will have 990 : 9 = 110 bottles.\r\n" );
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document.write( "The line of their centers is y = 4.5 cm.\r\n" );
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document.write( "The centers of the circles form a grid of equilateral triangles with the side length of 9 cm.\r\n" );
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document.write( "When the side of an equilateral triangle is a = 9 cm, its height/altitude is \r\n" );
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document.write( " h =  =  = 7.79423 cm (rounded up).\r\n" );
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document.write( "So, the second row of circles is 7.79423 cm above the first horizontal line y = 4.5 cm.\r\n" );
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document.write( "Thus, the equation of the 2nd horizontal line of centers is y = 4.5 + 7.79423, or y = 12.29423 cm.\r\n" );
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document.write( "The most upper horizontal line of circles is not closer than 4.5 cm to the upper bound y = 990 cm.\r\n" );
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document.write( "Therefore, if the total number of horizontal lines of centers is n, we can write this inequality\r\n" );
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document.write( " 4.5 + (n-1)*7.79423 + 4.5 <= 990 cm,\r\n" );
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document.write( "or\r\n" );
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document.write( " (n-1)*7.79423 <= 990 - 4.5 - 4.5 = 990 - 9 = 981 cm.\r\n" );
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document.write( "S0, the number of lines is about (is not more than)\r\n" );
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document.write( " n <=  + 1 = 125.8623366 + 1.\r\n" );
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document.write( "Rounding to the closest lesser integer number, we see that\r\n" );
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document.write( " n <= 126.\r\n" );
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document.write( "We see that there is the room for 126 lines of centers, but there is no room for 127 lines.\r\n" );
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document.write( "The number of bottles / (circles) in lines is altering 110, 109, 110, 109, . . . , 110, 109, 110, 109.\r\n" );
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document.write( "So, this sequence has 63 lines with 110 bottles and 63 lines with 109 bottles.\r\n" );
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document.write( "Hence, the total maximum possible number of bottles is 63*110 + 63*109 = 13797.\r\n" );
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document.write( "ANSWER. The total maximum possible number of bottles in the box is 13,797.\r\n" );
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document.write( "Solved.\r
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