document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120621>Question 1120621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The store sells sixty CD players everyday for $80 each. Each $4 increase in price reduces their sales by 2 playerss. Find the maximum daily income from sales and the price they should charge to get it.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #736279 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=736279\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the store sells 60 CD players every day for $80 each.\r<BR>\n" );
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document.write( "each $4 increase in price reduces their sales by 2 players.\r<BR>\n" );
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document.write( "find the maximum daily income from sales and the price they should charge to get it.\r<BR>\n" );
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document.write( "the total income each day is equal to 80 * 60 with the price being 80 and the quantity sold being 60.\r<BR>\n" );
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document.write( "let x equal the change in the price.\r<BR>\n" );
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document.write( "when the price goes up 4, the quantity goes down 2, therefore the change in the quantity is half the change in the price.\r<BR>\n" );
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document.write( "the equation to model this is:\r<BR>\n" );
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document.write( "total income = (80 + x) * (60 - x/2)\r<BR>\n" );
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document.write( "when x = 0, the total income is equal to 80 * 60 = 4800\r<BR>\n" );
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document.write( "when x = -4, the total income is equal to 76 * 62 = 4712\r<BR>\n" );
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document.write( "when x = 4, the total income is equal to 84 * 58 = 4872\r<BR>\n" );
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document.write( "this equation can be graphed by setting y = total income.\r<BR>\n" );
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document.write( "the equation becomes y = (80 + x) * (60 - x/2)\r<BR>\n" );
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document.write( "the first graph shows the maximum total income which we will derive algebraically further down.\r<BR>\n" );
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document.write( "the second graph shows the total income when the price goes down 4.\r<BR>\n" );
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document.write( "the third graph shows the total income when the price goes up 4.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/072905.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/072906.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/072907.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the equation is y = (80 + x) * (60 - x/2)\r<BR>\n" );
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document.write( "perform the indicated multiplication and the equation becomes:\r<BR>\n" );
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document.write( "y = 4800 - 40 * x + 60 * x - x^2/2\r<BR>\n" );
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document.write( "combine like terms and rearrange the terms in descending order of degree to get:\r<BR>\n" );
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document.write( "y = -x^2/2 + 20 * x + 4800\r<BR>\n" );
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document.write( "x^2 can be shown at .5 * x^2, therefore the equation becomes:\r<BR>\n" );
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document.write( "y = -.5 * x^2 + 20 * x + 4800\r<BR>\n" );
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document.write( "this equation is in standard form of y = ax^2 + bx + c, where:\r<BR>\n" );
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document.write( "a = -.5<BR>\n" );
document.write( "b = 20<BR>\n" );
document.write( "c = 4800\r<BR>\n" );
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document.write( "the maximum value of this equation is when x = -b/2a.\r<BR>\n" );
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document.write( "-b/2a becomes -20 / (2 * -.5) which becomes -20 / -1 which becomes 20.\r<BR>\n" );
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document.write( "the equation is at its maximum value when x = 20.\r<BR>\n" );
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document.write( "when x = 20, y = -.5 * 20^2 + 20 * 20 + 4800 which results in y = 5000.\r<BR>\n" );
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document.write( "this means that total income, represented by y, is equal to a maximum value of 5000 when the price is 20 dollars above 80.\r<BR>\n" );
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document.write( "the price is 80 + 20 = 100 and the quantity sold is 60 - 10 = 50.\r<BR>\n" );
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document.write( "100 * 50 = 5000.\r<BR>\n" );
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document.write( "remember the equation.\r<BR>\n" );
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document.write( "total income = (80 + x) * (60 - x/2)\r<BR>\n" );
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document.write( "x represents the increase in dollars.\r<BR>\n" );
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document.write( "if the price goes up 4, then x = 4.<BR>\n" );
document.write( "if the price does down 4, then x = -4.\r<BR>\n" );
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document.write( "for every increase of 4 dollars, the quantity goes down by 2.\r<BR>\n" );
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document.write( "for every decrease of 4 dollars, the quantity goes up 2.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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