document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203126>Question 1203126</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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<BR>\n" );
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document.write( "a) Determine the price of a blender that will maximize revenue for the month.\r<BR>\n" );
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document.write( "b) Approximately how many blenders will be sold to reach revenue of $1100?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838443 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=838443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> revenue = price * quantity sold<BR>\n" );
document.write( "when price = 70, quantity = 12 making revenue = 70 * 12 = 840.<BR>\n" );
document.write( "when the price decreases by 5 dollars, the quantity sold increased by 3 dollars.<BR>\n" );
document.write( "let x equal the number of times the price decreases and the quantity increases.<BR>\n" );
document.write( "the equation becomes revenue = (70 - 5x) * (12 + 3x)<BR>\n" );
document.write( "when x = 0, revenue = 70 * 12 = 840<BR>\n" );
document.write( "when x = 1, revenue = 65 * 15 = 975<BR>\n" );
document.write( "when x = 2, revenue = 60 * 18 = 1080<BR>\n" );
document.write( "when x = 3, revenue = 55 * 21 = 1155\r<BR>\n" );
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document.write( "your revenue equation is y = (70 - 5x) * (12 + 3x).<BR>\n" );
document.write( "simpiify to get y = 840 + 150x - 15x^2.<BR>\n" );
document.write( "rearrange by descending order of degree to get -15x^2 + 150x + 840.\r<BR>\n" );
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document.write( "for the revenue to be equal to 1100, the equaton becomes:<BR>\n" );
document.write( "-15x^2 + 150x + 840 = 1100<BR>\n" );
document.write( "subtract 1100 from both sides of the equation to get -15x^2 + 150x + 840 -1100 = 0.<BR>\n" );
document.write( "combine like terms to get -15x^2 + 150x - 260 = 0.<BR>\n" );
document.write( "multiply both sides of the equation by -1 to get  15x^2 - 150x + 260 = 0.<BR>\n" );
document.write( "factor the equation to get x = 7.7688746209727 or x = 2.2311253790273.\r<BR>\n" );
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document.write( "your original equation is y = -15x^2 + 150x + 840<BR>\n" );
document.write( "the general form of this equation is y = ax^2 + bx + c.<BR>\n" );
document.write( "y will be maximum when x = -b/(2a) = -150/-30 = 5.<BR>\n" );
document.write( "when x = 5, y = -15*5^2 + 150*5 + 840 = 1215.<BR>\n" );
document.write( "that's the maximum revenue.<BR>\n" );
document.write( "it occurs when the price is 70 - 5*5 = 45 and the number of blenders sold is 12 + 5*3 = 27.<BR>\n" );
document.write( "45 * 27 = max revenue of 1215.<BR>\n" );
document.write( "this is shown on the graph at the point (5,1215).<BR>\n" );
document.write( "the value of x is 5 for the maximum revenue.<BR>\n" );
document.write( "this means 5 increments of 5 dollars less for the price and 5 increments of 3 units more for the quantity.\r<BR>\n" );
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document.write( "the number of increments is not an integer when you want the revenue to be exactly 1100.<BR>\n" );
document.write( "the values of x that allow a revenue of at or above 1100 are 2.2311253790273 <= x <= 7.7688746209727.<BR>\n" );
document.write( "if you want to make the increments integer, then you would choose x = 3 to 7.<BR>\n" );
document.write( "that would make the revenue greater than 1100.\r<BR>\n" );
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document.write( "here are two displays of the graph and one display of the spreadshee i used to make the calculations.\r<BR>\n" );
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document.write( "i'll be available to answer any questions you might have.<BR>\n" );
document.write( "theo </SPAN>\n" );
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