document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1056801>Question 1056801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer can sell headphones at a price of 140- .01x dollars each. It costs 40x + 15,000 dollars to produce all of them. How many headphones should be produced to maximize profit? (application of derivative) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #672086 by </B><A HREF=/tutors/aboutme.mpl?userid=solve_for_x><B>solve_for_x(190)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solve_for_x><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=672086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Profit is the difference between revenue and cost.\r<BR>\n" );
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document.write( "With a price of p = 140 - 0.01x, the sale of x headphones gives revenue of:\r<BR>\n" );
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document.write( "R(x) = px = (140 - 0.01x)x = 140x - 0.01x^2\r<BR>\n" );
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document.write( "The costs are:\r<BR>\n" );
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document.write( "C(x) = 40x + 15000\r<BR>\n" );
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document.write( "The profit is then:\r<BR>\n" );
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document.write( "P(x) = R(x) - C(x) = (140x - 0.01x^2) - (40x + 15000)\r<BR>\n" );
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document.write( "P(x) = -0.01x^2 + 140x - 40x - 15000\r<BR>\n" );
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document.write( "P(x) = -0.01x^2 + 100x - 15000\r<BR>\n" );
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document.write( "Taking the derivative of P(x) gives:\r<BR>\n" );
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document.write( "P'(x) = -0.01(2)x + 100\r<BR>\n" );
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document.write( "P'(x) = -0.02x + 100\r<BR>\n" );
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document.write( "Setting the derivative equal to 0 and solving for x then gives:\r<BR>\n" );
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document.write( "P'(x) = -0.02x + 100 = 0\r<BR>\n" );
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document.write( "0.02x = 100\r<BR>\n" );
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document.write( "x = 100 / 0.02\r<BR>\n" );
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document.write( "x = 5000\r<BR>\n" );
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document.write( "The maximum profit corresponds to the production of 5000 headphones. </SPAN>\n" );
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