document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=42585>Question 42585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Cant remember how to solve this can anyone help<BR>\n" );
document.write( "4000exp(-400t)-50000exp(-1000t)-64000exp(-1600t)=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27767 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=27767\">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=4000e%5E%28-400t%29-50000e%5E%28-1000t%29-64000e%5E%28-1600t%29=0 ALIGN=MIDDLE  ALT=\"4000e%5E%28-400t%29-50000e%5E%28-1000t%29-64000e%5E%28-1600t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Factor out the common factors, which would be 4000 and e raised to the lowest power of the variable, which is e^(-1600t).  Remember that when you factor out the common factor, you must SUBTRACT exponents:\r<BR>\n" );
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document.write( "The common factor will be:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000+e%5E%28-1600t%29+ ALIGN=MIDDLE  ALT=\"2000+e%5E%28-1600t%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000+e%5E%28-1600t%29%2A+%282e%5E%28-400t%2B1600t%29+-+25e%5E%28-1000t%2B1600t%29+-+32+%29=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000+e%5E%28-1600t%29%2A+%282e%5E%281200t%29+-+25e%5E%28600t%29+-+32+%29=+0 ALIGN=MIDDLE  ALT=\"2000+e%5E%28-1600t%29%2A+%282e%5E%281200t%29+-+25e%5E%28600t%29+-+32+%29=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first factor can never equal zero, so you need to solve:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282e%5E%281200t%29+-+25e%5E%28600t%29+-+32+%29=+0 ALIGN=MIDDLE  ALT=\"%282e%5E%281200t%29+-+25e%5E%28600t%29+-+32+%29=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substitution may be helpful here, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+e%5E%28600t%29 ALIGN=MIDDLE  ALT=\"u+=+e%5E%28600t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2+=+e%5E%281200t%29 ALIGN=MIDDLE  ALT=\"u%5E2+=+e%5E%281200t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%5E2+-+25u+-+32=0 ALIGN=MIDDLE  ALT=\"2u%5E2+-+25u+-+32=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and this does NOT factor, so you'll have to use the quadratic formula to solve it.  Then substitute the value back and solve for t. \r<BR>\n" );
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document.write( "Solving with quadratic formula, you get  <BR>\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=2x%5E2%2B-25x%2B-32+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-25x%2B-32+=+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=%28-25%29%5E2-4%2A2%2A-32=881 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-25%29%5E2-4%2A2%2A-32=881\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  Discriminant d=881 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--25%2B-sqrt%28+881+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--25%2B-sqrt%28+881+%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-%28-25%29%2Bsqrt%28+881+%29%29%2F2%5C2+=+13.6704110398279 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-25%29%2Bsqrt%28+881+%29%29%2F2%5C2+=+13.6704110398279\"   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-%28-25%29-sqrt%28+881+%29%29%2F2%5C2+=+-1.17041103982791 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-25%29-sqrt%28+881+%29%29%2F2%5C2+=+-1.17041103982791\"   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=2x%5E2%2B-25x%2B-32 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-25x%2B-32\"   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=2x%5E2%2B-25x%2B-32+=+%28x-13.6704110398279%29%2A%28x--1.17041103982791%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-25x%2B-32+=+%28x-13.6704110398279%29%2A%28x--1.17041103982791%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 13.6704110398279, -1.17041103982791.\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+2%2Ax%5E2%2B-25%2Ax%2B-32+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-25%2Ax%2B-32+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Now, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+e%5E%28600t%29 ALIGN=MIDDLE  ALT=\"u+=+e%5E%28600t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, to solve for t, will a decimal approximation be good enough?  Otherwise it gets really hairy!!  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.67+=+e%5E%28600t%29 ALIGN=MIDDLE  ALT=\"13.67+=+e%5E%28600t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >   or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.17+=+e%5E%28600t%29 ALIGN=MIDDLE  ALT=\"-1.17+=+e%5E%28600t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The second answer has no solution, since e raised to a power cannot equal a negative.  This means that the first equation is the only one that has a solution.\r<BR>\n" );
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document.write( "You will need to take the ln of each side of this equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln+%2813.67%29+=+ln%28e%5E%28600t%29%29+=+600t+ ALIGN=MIDDLE  ALT=\"+ln+%2813.67%29+=+ln%28e%5E%28600t%29%29+=+600t+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide both sides by 600:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28ln+13.67%29%2F600 ALIGN=MIDDLE  ALT=\"t=%28ln+13.67%29%2F600\"   DISABLED_style= \"max-width:90%; height:auto;\" > or approximately 0.00436  \r<BR>\n" );
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document.write( "If you want to check this answer, you can always confirm the solution to this with a graphing calculator, using the \"zeros\" or the \"root\" method!!\r<BR>\n" );
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document.write( "By the way, the exact value of t would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=+%28ln%28%2825%2Bsqrt%28881%29%29%2F4%29%29%2F600+ ALIGN=MIDDLE  ALT=\"t=+%28ln%28%2825%2Bsqrt%28881%29%29%2F4%29%29%2F600+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Check it out, and see if the decimal approximation of this rounds off to 0.0043587229.\r<BR>\n" );
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document.write( "R^2 at SCC  </SPAN>\n" );
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