document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1140030>Question 1140030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) A manufacture has been selling 1650 television sets a week at $540 each. A market survey indicates that for each $13 rebate offered to a buyer, the number of sets sold will increase by 130 per week. \r<BR>\n" );
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document.write( "a) Find the function representing the demand p(x), where x is the number of the television sets sold per week and p(x) is the corresponding price. <BR>\n" );
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document.write( "b) How large rebate should the company offer to a buyer, in order to maximize its revenue? <BR>\n" );
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document.write( "c) If the weekly cost function is 148500+180x, how should it set the size of the rebate to maximize its profit?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760516 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=760516\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "when your demand is 1650 and the price is 540, your revenue is equal to demand * price = 1650 * 540 = 891,000.\r<BR>\n" );
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document.write( "your cost is equal to 148,500 + 180 * the demand which becomes 148,500 + 180 * 1650 = 445,500.\r<BR>\n" );
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document.write( "your profit is then equal to revenue minus cost = 891,000 - 445,500 = 445,500.\r<BR>\n" );
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document.write( "you are given that, when the price goes down by 13 dollars a unit, the demand goes up by 130 units.\r<BR>\n" );
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document.write( "the equation for that is revenue = (1650 + 130 * x) * (540 - 13 * x).\r<BR>\n" );
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document.write( "the demand is (1650 + 130 * x) and the price per unit is (540 - 13 * x).\r<BR>\n" );
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document.write( "x does not represent the demand, nor does it represent the revenue.\r<BR>\n" );
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document.write( "130 * x represents the amount of change in the demand and 13 * x represents the amount of change in the price per unit.\r<BR>\n" );
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document.write( "when x = 0, the amount of change in the demand is 0 and the amount of change in the price is 0.\r<BR>\n" );
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document.write( "when x = 1, the amount of change in the demand is equal to plus 130 and the amount of change in the price is equal to minus 13 dollars.\r<BR>\n" );
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document.write( "when x = 2, the amount of change in the demand is equal to plus 260 and the amount of change in the price is equal to minus 26 dollars.\r<BR>\n" );
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document.write( "as such, x represents the number of increments of change in the demand and in the corresponding price.\r<BR>\n" );
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document.write( "it does not represent the demand itself.\r<BR>\n" );
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document.write( "this means that your cost equation, shown as 148500 + 180 * x, needs to be changed because the demand is represented by 1650 + 130 * x, and not x.\r<BR>\n" );
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document.write( "the cost equation becomes cost = 148,500 + 180 * (1650 + 130 * x).\r<BR>\n" );
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document.write( "you have a revenue equation equal to (1650 + 130 * x) * (540 - 130 * x).\r<BR>\n" );
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document.write( "you have a cost equation equal to 148,500 + 180 * (1650 + 130 * x).\r<BR>\n" );
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document.write( "since profit is equal to revenue minus cost, then your profit equation becomes:\r<BR>\n" );
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document.write( "profit = (1650 + 130 * x) * (540 - 130 * x) - (148,500 + 180 * (1650 + 130 * x)).\r<BR>\n" );
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document.write( "the revenue equation can be simplified to:\r<BR>\n" );
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document.write( "revenue = -1690x^2 + 48750x + 891000, which is a quadratic equation.\r<BR>\n" );
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document.write( "since it is in standard form, you get:\r<BR>\n" );
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document.write( "a = -1690<BR>\n" );
document.write( "b = 48750<BR>\n" );
document.write( "c = 891000\r<BR>\n" );
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document.write( "the maximum revenue occurs when x = -b/2a.\r<BR>\n" );
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document.write( "solve for x = -b/2a to get x = 14.42307692.\r<BR>\n" );
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document.write( "the maximum revenue is determined when you replace x in the revenue equation with 14.42307692.\r<BR>\n" );
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document.write( "this gets you max revenue of 1,242,562.5 dollars.\r<BR>\n" );
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document.write( "the answer to question a is:\r<BR>\n" );
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document.write( "the revenue function is revenue = (1650 + 130 * x) * (540 - 13 * x)\r<BR>\n" );
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document.write( "the demand is represented by (165 + 130 * x)\r<BR>\n" );
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document.write( "the price is represented by (540 - 13 * x)\r<BR>\n" );
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document.write( "x represents the number of increments of change required.\r<BR>\n" );
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document.write( "the answer to question b is:\r<BR>\n" );
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document.write( "revenue is maximized when the rebate is equal to 13 * 14.42307692 = 187.5 dollars.\r<BR>\n" );
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document.write( "when x = 14.423076972, the demand is equal to 1650 + 130 * 14.42307692 = 3525 and the price is equal to 540 - 187.5 = 352.50 dollars.\r<BR>\n" );
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document.write( "the answer to question c is:\r<BR>\n" );
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document.write( "the profit is maximized when the rebate is equal to 13 * 7.5 = 97.5 dollars.\r<BR>\n" );
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document.write( "when x = 7.5, the demand is equal to 1650 + 130 * 7.5 = 2625 and the price is equal to 540 - 13 * 7.5 = 442.5 dollars.\r<BR>\n" );
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document.write( "the graph of the revenue and profit functions are shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042811.jpg\" alt=\"$$$\">\r<BR>\n" );
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document.write( "the display of the excel spreadsheet calculations are shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042812.jpg\" alt=$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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