document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104108>Question 1104108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The demand curve for a product is given by  q=800-7p^2 ,  where  p is the price. Find the price that maximizes revenue for sales of this product.<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 #718838 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=718838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "demand equation: q=800-7p^2<BR>\n" );
document.write( "q = number of items demanded (aka number of items willing to be bought)<BR>\n" );
document.write( "p = price per item in dollars\r<BR>\n" );
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document.write( "Revenue = (price per item)*(number of items bought)<BR>\n" );
document.write( "R = (p)*(q)<BR>\n" );
document.write( "R = p*(800-7p^2)<BR>\n" );
document.write( "R = p*800+p*(-7p^2)<BR>\n" );
document.write( "R = 800p-7p^2<BR>\n" );
document.write( "R = -7p^2+800p\r<BR>\n" );
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document.write( "Let f(x) = -7x^2+800x. Finding the vertex of f(x) will lead to the max value of R(p)\r<BR>\n" );
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document.write( "In the case of f(x) = -7x^2+800x, it is in the form f(x) = ax^2+bx+c. So a = -7, b = 800 and c = 0\r<BR>\n" );
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document.write( "Vertex = (h,k)\r<BR>\n" );
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document.write( "Use this formula <BR>\n" );
document.write( "h = -b/(2*a)<BR>\n" );
document.write( "to find the x coordinate of the vertex\r<BR>\n" );
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document.write( "h = -b/(2*a)<BR>\n" );
document.write( "h = -800/(2*(-7)) <<---- plugging in a = -7 and b = 800<BR>\n" );
document.write( "h = -800/(-14)<BR>\n" );
document.write( "h = 800/14<BR>\n" );
document.write( "h = 400/7<BR>\n" );
document.write( "h = 57.1428571428571 <<---- this value is approximate<BR>\n" );
document.write( "h = 57.14 <<---- rounding to 2 decimal places (aka to the nearest penny)\r<BR>\n" );
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document.write( "The x coordinate of the vertex is roughly 57.14. Since I replaced p with x, this indicates that the x coordinate of the vertex corresponds to the p coordinate of (p, R(p))\r<BR>\n" );
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document.write( "So the max revenue will occur when the price per item is <font color=red>$57.14</font>\r<BR>\n" );
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document.write( "Extra Info:\r<BR>\n" );
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document.write( "Plug the value of h into the f(x) function to get the y coordinate of the vertex<BR>\n" );
document.write( "This will give the max revenue<BR>\n" );
document.write( "k = f(h)<BR>\n" );
document.write( "k = -7(57.14)^2+800(57.14)<BR>\n" );
document.write( "k = 22,857.1428<BR>\n" );
document.write( "k = 22,857.14<BR>\n" );
document.write( "The y coordinate of the vertex is 22,857.14 indicating the largest revenue possible is $22,857.14 (which happens when the price per item is set at $57.14)\r<BR>\n" );
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