document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=59633>Question 59633</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780073268620/algebrahomeworkh>Elementry and Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> The daily profit of a company P is related to the number of employees (x) working on that day and is represented by the equation P = - 25x2 + 300x. <BR>\n" );
document.write( "What number of employees will maximize the profit and what will be the maximum profit?   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #40911 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=40911\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-+25x2+%2B+300x ALIGN=MIDDLE  ALT=\"P+=+-+25x2+%2B+300x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The negative sign in front of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term means the graph will <BR>\n" );
document.write( "rise, then peak, then fall.<BR>\n" );
document.write( "There should be two points where the profit,P, is zero.<BR>\n" );
document.write( "One point is where x=0, which means no employees. That<BR>\n" );
document.write( "makes sense, doesn't it?<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-+25%2A0%5E2+%2B+300%2A0 ALIGN=MIDDLE  ALT=\"0+=+-+25%2A0%5E2+%2B+300%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the point (0, 0)<BR>\n" );
document.write( "The other zero-profit point is where the graph rises and then falls<BR>\n" );
document.write( "back to the x-axis, where P, the vertical axis, is zero.<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%2A%28-25x+%2B+300%29 ALIGN=MIDDLE  ALT=\"0+=+x%2A%28-25x+%2B+300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is true when x=0, as I just showed, and also when (-25x + 300) = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-25x+%2B+300+=+0 ALIGN=MIDDLE  ALT=\"-25x+%2B+300+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-25x+=+-300 ALIGN=MIDDLE  ALT=\"-25x+=+-300\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+12 ALIGN=MIDDLE  ALT=\"x+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, with 12 employees, the profit is back to zero again.<BR>\n" );
document.write( "This is the point (12, 0)<BR>\n" );
document.write( "The vertex, or peak of the graph is mid-way between these two<BR>\n" );
document.write( "x-values, x=0 and x=12<BR>\n" );
document.write( "That means x = 6 should give us the max profit<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+-25%2A6%5E2+%2B+300%2A6 ALIGN=MIDDLE  ALT=\"P+=+-25%2A6%5E2+%2B+300%2A6\"   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%5Bmax%5D+=+900 ALIGN=MIDDLE  ALT=\"P%5Bmax%5D+=+900\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is at the point (6, 900)<BR>\n" );
document.write( "We can check that, with 6 employees, the profit is maximum, and it<BR>\n" );
document.write( "is 900. Try putting x = 5.9 and x = 6.1 into the equation and see<BR>\n" );
document.write( "what you get for P. I did and I got<BR>\n" );
document.write( "P(5.9) = 899.75<BR>\n" );
document.write( "P(6.1) = 899.75<BR>\n" );
document.write( "So the profit drops slightly on either side of x = 6. That is a strong<BR>\n" );
document.write( "indication that (6, 900) is a maximum.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );