document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316385>Question 316385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> S= -3.5n^2+42n+45. S=number of thousands of dollars of sales in week n.<BR>\n" );
document.write( "When do you expect the sales to peak?<BR>\n" );
document.write( "What is the largest value for the sales during the week?<BR>\n" );
document.write( "During what week do we expect the sales to drop to zero? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226369 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=226369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> S= -3.5n^2+42n+45. S=number of thousands of dollars of sales in week n.<BR>\n" );
document.write( "When do you expect the sales to peak?<BR>\n" );
document.write( "vertex gives you the max.<BR>\n" );
document.write( "axis of symmetry:<BR>\n" );
document.write( "n = -b/(2a) = -42/(2*(-3.5)) = -42/-7 = 6<BR>\n" );
document.write( "It will peak at n=6 or the sixth week<BR>\n" );
document.write( ".<BR>\n" );
document.write( "What is the largest value for the sales during the week?<BR>\n" );
document.write( "S= -3.5n^2+42n+45<BR>\n" );
document.write( "set n=6<BR>\n" );
document.write( "S= -3.5*6^2+42(6)+45<BR>\n" );
document.write( "S= -3.5*36+252+45<BR>\n" );
document.write( "S= -126+252+45<BR>\n" );
document.write( "s= 171 (thousands of dollars)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "During what week do we expect the sales to drop to zero?<BR>\n" );
document.write( "set S = 0 solve for n:<BR>\n" );
document.write( "S= -3.5n^2+42n+45<BR>\n" );
document.write( "0= -3.5n^2+42n+45<BR>\n" );
document.write( "Applying the quadratic formula yields:<BR>\n" );
document.write( "n = {-0.99, 12.99}<BR>\n" );
document.write( "We can toss out the negative answer leaving:<BR>\n" );
document.write( "n = 12.99 weeks<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Detail of quadratic follows:<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=an%5E2%2Bbn%2Bc=0 ALIGN=MIDDLE  ALT=\"an%5E2%2Bbn%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.5n%5E2%2B42n%2B45+=+0 ALIGN=MIDDLE  ALT=\"-3.5n%5E2%2B42n%2B45+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2842%29%5E2-4%2A-3.5%2A45=2394 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2842%29%5E2-4%2A-3.5%2A45=2394\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=2394 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-42%2B-sqrt%28+2394+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-42%2B-sqrt%28+2394+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B1%5D+=+%28-%2842%29%2Bsqrt%28+2394+%29%29%2F2%5C-3.5+=+-0.98978847012861 ALIGN=MIDDLE  ALT=\"n%5B1%5D+=+%28-%2842%29%2Bsqrt%28+2394+%29%29%2F2%5C-3.5+=+-0.98978847012861\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B2%5D+=+%28-%2842%29-sqrt%28+2394+%29%29%2F2%5C-3.5+=+12.9897884701286 ALIGN=MIDDLE  ALT=\"n%5B2%5D+=+%28-%2842%29-sqrt%28+2394+%29%29%2F2%5C-3.5+=+12.9897884701286\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.5n%5E2%2B42n%2B45 ALIGN=MIDDLE  ALT=\"-3.5n%5E2%2B42n%2B45\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.5n%5E2%2B42n%2B45+=+-3.5%28n--0.98978847012861%29%2A%28n-12.9897884701286%29 ALIGN=MIDDLE  ALT=\"-3.5n%5E2%2B42n%2B45+=+-3.5%28n--0.98978847012861%29%2A%28n-12.9897884701286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.98978847012861, 12.9897884701286.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3.5%2Ax%5E2%2B42%2Ax%2B45+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-3.5%2Ax%5E2%2B42%2Ax%2B45+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );