document.write( "Question 1203126: At a price of $70 for a blender, Home Outfitters will sell 12 in one month. Market research has shown that for every $5 decrease in the price of a blender, they will be able to sell 3 more each month.\r
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Algebra.Com's Answer #838438 by ikleyn(52780)\"\" \"About 
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\n" ); document.write( "At a price of $70 for a blender, Home Outfitters will sell 12 in one month.
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\n" ); document.write( "(b) Approximately how many blenders will be sold to reach revenue of $1100?
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\n" ); document.write( "\n" ); document.write( "                    Solution for (a)\r
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document.write( "From the given information, we easily derive that 12+3n blenders\r\n" );
document.write( "can be sold at market at the price 70-5n dollars, where n is \"any\" integer number,\r\n" );
document.write( "meaning n steps of $5 decrease the initial price of $70.\r\n" );
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document.write( "Hence, the formula for revenue selling blenders at the price 70-5n dollars is\r\n" );
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document.write( "    R(70-5n) = (70-5n)*(12+3n) dollars.\r\n" );
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document.write( "We want to find the value of the argument 70-5n, which provides the maximum \r\n" );
document.write( "to the quadratic function (70-5n)*(12+3n).\r\n" );
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document.write( "For it, notice that the quadratic function (70-5n)*(12+3n), just being decomposed into\r\n" );
document.write( "the product of linear terms, has the zeroes (= x-intecepts) at\r\n" );
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document.write( "    \"70%2F5\" = 14  and  \"-12%2F3\" = -4.\r\n" );
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document.write( "Thus it has the maximum half-way between the x-intercepts.  This half-way value is \r\n" );
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document.write( "    \"n%5Boptimum%5D\" = \"%2814+%2B+%28-4%29%29%2F2\" = \"10%2F2\" = 5.\r\n" );
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document.write( "So, to get an optimum price, we need to make 5 steps decreasing the initial price of $70 by $5 each time.\r\n" );
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document.write( "Thus we get the optimum price of 70-5*5 = 70-25 = 45 dollars.\r\n" );
document.write( "At this price, 12+3n = 12+3*5 = 12+15 = 27 blenders can be solved,\r\n" );
document.write( "providing the maximum possible revenue of 45*27 = 1215 dollars.\r\n" );
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document.write( "Compare it with the revenue of 12*70 = 840 dollars, corresponding to the initial condition.\r\n" );
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document.write( "To answer (b), we should find integer values n that provide the closest values of (70-5n)*(12+3n) to 1100.\r\n" );
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document.write( "To get it, I used MS Excel and prepared this Table below\r\n" );
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document.write( " n	70-5n	12+3n	 product\r\n" );
document.write( "                        (70-5n)*(12+3n)\r\n" );
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document.write( " 1	 65	 15	  975\r\n" );
document.write( " 2	 60	 18	 1080    <<<---===\r\n" );
document.write( " 3	 55	 21	 1155\r\n" );
document.write( " 4	 50	 24	 1200\r\n" );
document.write( " 5	 45	 27	 1215\r\n" );
document.write( " 6	 40	 30	 1200\r\n" );
document.write( " 7	 35	 33	 1155\r\n" );
document.write( " 8	 30	 36	 1080    <<<---===\r\n" );
document.write( " 9	 25	 39	  975\r\n" );
document.write( "10	 20	 42	  840\r\n" );
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document.write( "The desired values of n are n= 2  and  n= 8.\r\n" );
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document.write( "The corresponding prices per blender are        70-5*2 = 60 dollars  or  70-5*8 = 30 dollars  (two possible values).\r\n" );
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document.write( "The corresponding amounts of blenders sold are  12+3*2 = 18          or  12+3*8 = 36          (two possible values).\r\n" );
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