document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=45497>Question 45497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need your help with this problem\r<BR>\n" );
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document.write( "A study done for management shows that the revenue, R, in dollars, as a function of number, x, of units produced can be modeled by <BR>\n" );
document.write( " R(x) = 10x - 0.001x^2 <BR>\n" );
document.write( "while the cost, C, in dollars, can be modeled by<BR>\n" );
document.write( "  C(x) = 5000 + 2x.  <BR>\n" );
document.write( "What will the maximum profit be, in dollars?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #30241 by </B><A HREF=/tutors/aboutme.mpl?userid=Fermat><B>Fermat(136)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fermat><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=Fermat><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=30241\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Profit = Revenue - Cost<BR>\n" );
document.write( "P(x) = R(x) - C(x)<BR>\n" );
document.write( "P(x) = 10x - 0.001x² - (5000 + 2x)<BR>\n" );
document.write( "P(x) = 8x - 0.001x² - 5000<BR>\n" );
document.write( "To find the maximumn, first work out the turning point(s), which are where when dP/dx = 0.<BR>\n" );
document.write( "Differentiating P(x),<BR>\n" );
document.write( "dP/dx = 8 - 0.002x<BR>\n" );
document.write( "When dP/dx = 0 then 8 - 0.002x = 0<BR>\n" );
document.write( "8 - 0.002x = 0<BR>\n" );
document.write( "8 = 0.002x<BR>\n" );
document.write( "x = 8/0.002<BR>\n" );
document.write( "x = 4000<BR>\n" );
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document.write( "There is thus a turning point at x = 4000.<BR>\n" );
document.write( "To find out whether it is a maximum or a minimum, take the 2nd derivative.<BR>\n" );
document.write( "This gives,<BR>\n" );
document.write( "d²P/dx² = -0.002<BR>\n" );
document.write( "The rule is:<BR>\n" );
document.write( "d²P/dx² > 0 means the turning point is a minimum<BR>\n" );
document.write( "d²P/dx² < 0 means the turning point is a maximum<BR>\n" );
document.write( "Since our d²P/dx² is -0.002 which is < 0, then our turning point is a maximum.<BR>\n" );
document.write( "Our maximum profit is at x = 4000 and is equal to,<BR>\n" );
document.write( "P(4000) = 8(4000) - 0.001(4000)² - 5000<BR>\n" );
document.write( "P = 32,000 - 16,000 - 5,000<BR>\n" );
document.write( "P = 11,000<BR>\n" );
document.write( "=========== </SPAN>\n" );
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