document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1132937>Question 1132937</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a florist sells 17000 bouquets per year at a cost of 9 dollars each. In the previous year they raised the price to 16 dollars, they only sold 10000 bouquets that year. Assuming the amount of bouquets sold is in a linear relationship with the cost, what price should the florist charge per bouquet to maximize their revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #750108 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=750108\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> when the price is 9, the quantity is 17000<BR>\n" );
document.write( "when the price is 16, the quantity is 10000\r<BR>\n" );
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document.write( "when the price goes up 7, the quantity goes down 7000.\r<BR>\n" );
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document.write( "7000 / 7 = 1000\r<BR>\n" );
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document.write( "for every unit that the price goes up,the quantity goes down 1000 units.\r<BR>\n" );
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document.write( "this is what they mean when they say the amount of bouquets sold is in a linear relationship with the cost.\r<BR>\n" );
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document.write( "that particular linear relationship can be shown as a straight line on a graph.\r<BR>\n" );
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document.write( "you have two possible equations.\r<BR>\n" );
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document.write( "the first one is where x is the number of units sold and y is the price.\r<BR>\n" );
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document.write( "the second one is where x is the price and y is the number of units sold.\r<BR>\n" );
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document.write( "here's what the equation and graph of the first one looks like.<BR>\n" );
document.write( "y represents the price and x represents the number of bouquets sold.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/011806.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "here's what the equation and graph of the second one looks like.<BR>\n" );
document.write( "y represents the number of bouquets sold and x represents the price.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/011807.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the revenue equation, however, is not a straight line.\r<BR>\n" );
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document.write( "the equation is revenue = price * quantity.\r<BR>\n" );
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document.write( "when the price is 9, the quantity is 17000 and the equation becomes revenue = 9 * 17000.\r<BR>\n" );
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document.write( "you can solve for revenue to get 153000 = 9 * 17000.\r<BR>\n" );
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document.write( "153000 is the revenue when the price is 9 and the quantity sold is 17000.\r<BR>\n" );
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document.write( "from the linear relationship between price and quantity, you can see that when the price goes up one unit, the quantity goes down 1000 units.\r<BR>\n" );
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document.write( "letting x be the unit, then the formula becomes:\r<BR>\n" );
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document.write( "y = (9 + x) * (17000 - 1000 * x)\r<BR>\n" );
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document.write( "that's a quadratic equation.\r<BR>\n" );
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document.write( "apply the distributive law to get:\r<BR>\n" );
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document.write( "y = (9 + x) * (17000 - 1000x) becomes y = 9 * 17000 - 9 * 1000x + x * 17000 - x * 1000x\r<BR>\n" );
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document.write( "combine like terms to get y = 153000 + 8000 * x - 1000 * x^2.\r<BR>\n" );
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document.write( "order the terms in descending order of degree to get y = -1000x^2 + 8000x + 153000.\r<BR>\n" );
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document.write( "in this format, you get:\r<BR>\n" );
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document.write( "a = coefficient of x^2 term = -1000<BR>\n" );
document.write( "b = coefficient of x term = 8000<BR>\n" );
document.write( "c = constant term = 153000\r<BR>\n" );
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document.write( "y will be at its maximum value when x = -b/(2a) = -8000 / -2000 = 4.\r<BR>\n" );
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document.write( "when x = 4, y will be equal to -1000 * 16 + 8000 * 4 + 153000 = 169000.\r<BR>\n" );
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document.write( "that will be the maximum revenue.\r<BR>\n" );
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document.write( "the graph of the equation (9 + x) * (17000 - 1000x) and the equation -1000x^2 + 8000x + 153000 will be identical as shown on the following graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src =\"http://theo.x10hosting.com/2019/011805.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the maximum point on the graph is when x = 4.\r<BR>\n" );
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document.write( "when x = 4, the price is 9 + 4 = 13.\r<BR>\n" );
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document.write( "when x = 4, the quantity is 17000 - 4000 = 13000.\r<BR>\n" );
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document.write( "revenue = price * quantity = 13 * 13000 = 169000.\r<BR>\n" );
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document.write( "the coordinate point of (4,169000) on the graph shows that.<BR>\n" );
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