document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036215>Question 1036215</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you go about finding when the percent of foreign ­born citizen was at its highest and when it was at its lowest percent since 1990?\r<BR>\n" );
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document.write( "The U.S. population that was foreign-born since 1900 can be modeled by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+P%28t%29+=+0.00006t%5E3+-+0.007t%5E2+%2B+0.05t+%2B+14+ ALIGN=MIDDLE  ALT=\"+P%28t%29+=+0.00006t%5E3+-+0.007t%5E2+%2B+0.05t+%2B+14+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where t = 0 in 1990.\r<BR>\n" );
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document.write( "Here is the graph of the polynomial function: http://postimg.org/image/hz7mo786z/ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650920 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=650920\">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%28t%29+=+0.00006t%5E3+-+0.007t%5E2+%2B+0.05t+%2B+14+ ALIGN=MIDDLE  ALT=\"+P%28t%29+=+0.00006t%5E3+-+0.007t%5E2+%2B+0.05t+%2B+14+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where t = 0 in 1990.<BR>\n" );
document.write( "--------<BR>\n" );
document.write( "Find the 1st derivative.<BR>\n" );
document.write( "P'(t) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00018t%5E2+-+0.014t+%2B+0.05+ ALIGN=MIDDLE  ALT=\"0.00018t%5E2+-+0.014t+%2B+0.05+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---<BR>\n" );
document.write( "Solve the quadratic for t, the times of local max and min.<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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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=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00018x%5E2%2B-0.014x%2B0.05+=+0 ALIGN=MIDDLE  ALT=\"0.00018x%5E2%2B-0.014x%2B0.05+=+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=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%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=%28-0.014%29%5E2-4%2A0.00018%2A0.05=0.00016 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-0.014%29%5E2-4%2A0.00018%2A0.05=0.00016\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=0.00016 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--0.014%2B-sqrt%28+0.00016+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--0.014%2B-sqrt%28+0.00016+%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=x%5B1%5D+=+%28-%28-0.014%29%2Bsqrt%28+0.00016+%29%29%2F2%5C0.00018+=+74.0253073352042 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-0.014%29%2Bsqrt%28+0.00016+%29%29%2F2%5C0.00018+=+74.0253073352042\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-0.014%29-sqrt%28+0.00016+%29%29%2F2%5C0.00018+=+3.75247044257356 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-0.014%29-sqrt%28+0.00016+%29%29%2F2%5C0.00018+=+3.75247044257356\"   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=0.00018x%5E2%2B-0.014x%2B0.05 ALIGN=MIDDLE  ALT=\"0.00018x%5E2%2B-0.014x%2B0.05\"   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=0.00018x%5E2%2B-0.014x%2B0.05+=+%28x-74.0253073352042%29%2A%28x-3.75247044257356%29 ALIGN=MIDDLE  ALT=\"0.00018x%5E2%2B-0.014x%2B0.05+=+%28x-74.0253073352042%29%2A%28x-3.75247044257356%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 74.0253073352042, 3.75247044257356.\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+0.00018%2Ax%5E2%2B-0.014%2Ax%2B0.05+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+0.00018%2Ax%5E2%2B-0.014%2Ax%2B0.05+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "==========<BR>\n" );
document.write( "t = x + 1900<BR>\n" );
document.write( "--> 1903 & 1974<BR>\n" );
document.write( "---------<BR>\n" );
document.write( "You can see on the graph the 1903 is the max and 1974 is the minimum.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "To determine whether it's a max or min,  find the 2nd derivative of P(t)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "P\"(t) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00036t+-+0.014+ ALIGN=MIDDLE  ALT=\"0.00036t+-+0.014+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "P\"(3) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.00036t+-+0.014+ ALIGN=MIDDLE  ALT=\"0.00036t+-+0.014+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -0.01292<BR>\n" );
document.write( "< 0 --> maximum<BR>\n" );
document.write( "---<BR>\n" );
document.write( "P\"(74) = 0.01264<BR>\n" );
document.write( "> 0 --> minimum </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );