document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=23275>Question 23275</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Maximum profit using the quadratic equations, functions, inequalities and their graphs.\r<BR>\n" );
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document.write( "A chain store manager has been told by the main office that daily profit, P, is related to the number of clerks working that day, x, according to the equations P = -25x^2 + 300x.  What number of clerks will maximize the profit, and what is the maximum possible profit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #12018 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=12018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Maximise: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-25x%5E2+%2B+300x ALIGN=MIDDLE  ALT=\"P+=+-25x%5E2+%2B+300x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is the equation of a parabola that opens downwards (coefficient of x^2 is negative) so the maximum value of P (the dependent variable) will be found at the parabola's vertex. The x-coordinate of the vertex is given by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F2a ALIGN=MIDDLE  ALT=\"x+=+-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and so the maximum value of P will be found at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F2a ALIGN=MIDDLE  ALT=\"x+=+-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Your equation is already in the standard form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"P+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a = -25, b = 300, c = 0) so we can find the x-coordinate at which P will be the maximum.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%28300%29%2F2%28-25%29 ALIGN=MIDDLE  ALT=\"x+=+-%28300%29%2F2%28-25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-300%2F%28-50%29 ALIGN=MIDDLE  ALT=\"x+=+-300%2F%28-50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6 ALIGN=MIDDLE  ALT=\"x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To maximise profits, the manager should employ 6 clerks.\r<BR>\n" );
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document.write( "The maximum profit can be found by substituting 6 for x in the original equation for P.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-25%286%29%5E2+%2B+300%286%29 ALIGN=MIDDLE  ALT=\"P+=+-25%286%29%5E2+%2B+300%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-25%2836%29+%2B+1800 ALIGN=MIDDLE  ALT=\"P+=+-25%2836%29+%2B+1800\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-900+%2B+1800 ALIGN=MIDDLE  ALT=\"P+=+-900+%2B+1800\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+900 ALIGN=MIDDLE  ALT=\"P+=+900\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Maximum profit is 900 (dollars ?)<BR>\n" );
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