document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=133462>Question 133462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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 equation 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 #97634 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=97634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the graph of this relationship is a parabola.  In this case a = -25, b = 300, and c = 0.\r<BR>\n" );
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document.write( "Since you put this question in the Algebra: Quadratic Equations section, I'll show you the algebra method to solve it.  If you actually need the calculus method, write back and I'll show you that way.\r<BR>\n" );
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document.write( "A parabola with a lead coefficient that is less than zero opens downward, hence the vertex of the parabola is a maximum point.  The x-coordinate of the vertex of a parabola of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F2a ALIGN=MIDDLE  ALT=\"%28-b%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the y-coordinate is just the value of the function evaluated at the x-coordinate.\r<BR>\n" );
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document.write( "For your function, the vertex is at the point (x,y) where\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28green%28-300%29%29%2F2%28green%28-25%29%29=red%286%29 ALIGN=MIDDLE  ALT=\"x=%28green%28-300%29%29%2F2%28green%28-25%29%29=red%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=f%28red%286%29%29=-25%286%29%5E2%2B300%286%29=-900%2B1800=900 ALIGN=MIDDLE  ALT=\"y=f%28red%286%29%29=-25%286%29%5E2%2B300%286%29=-900%2B1800=900\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, the maximum profit will be achieved when 6 clerks are working, and that profit is 900 (anybody's guess as to whether that is dollars, thousands of dollars, euros, yen, or some other currency)\r<BR>\n" );
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document.write( "Following is a graphical illustration of the situation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C-1%2C13%2C-10%2C1000%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28600%2C600%2C-1%2C13%2C-10%2C1000%2C-25x%5E2%2B300x%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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