document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=214806>Question 214806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the problem. Please show your work.\r<BR>\n" );
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document.write( "1. The population of a colony of bacteria is decaying exponentially according to the function below, where t is the time in minutes. How many minutes will it take for the population of the colony to drop to 50?\r<BR>\n" );
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document.write( "B(t)=2000*e^(-.03*t)\r<BR>\n" );
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document.write( "e=2.718<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #162268 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=162268\">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=B%28t%29=2000%2Ae%5E%28-.03%2At%29 ALIGN=MIDDLE  ALT=\"B%28t%29=2000%2Ae%5E%28-.03%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Substitute 50 for B(t) to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+=+2000%2Ae%5E%28-.03%2At%29 ALIGN=MIDDLE  ALT=\"50+=+2000%2Ae%5E%28-.03%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Take the natural log of both sides to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2850%29+=+ln%282000%2Ae%5E%28-.03%2At%29%29 ALIGN=MIDDLE  ALT=\"ln%2850%29+=+ln%282000%2Ae%5E%28-.03%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "rules of logarithms apply here.<BR>\n" );
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document.write( "first rule:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28a%2Ab%29+=+ln%28a%29+%2B+ln%28b%29 ALIGN=MIDDLE  ALT=\"ln%28a%2Ab%29+=+ln%28a%29+%2B+ln%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "second rule:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28a%5Eb%29+=+b%2Aln%28a%29 ALIGN=MIDDLE  ALT=\"ln%28a%5Eb%29+=+b%2Aln%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "applying the first rule,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282000%2Ae%5E%28-.03%2At%29%29 ALIGN=MIDDLE  ALT=\"ln%282000%2Ae%5E%28-.03%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282000%29+%2B+ln%28e%5E%28-.03t%29%29 ALIGN=MIDDLE  ALT=\"ln%282000%29+%2B+ln%28e%5E%28-.03t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "applying the second rule,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%28-.03t%29%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%28-.03t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-.03t%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"-.03t%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "putting this all together, your equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2850%29+=+ln%282000%29+-+.03t%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%2850%29+=+ln%282000%29+-+.03t%2Aln%28e%29\"   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=ln%28e%29 ALIGN=MIDDLE  ALT=\"ln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals 1, so your equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2850%29+=+ln%282000%29+-+.03t ALIGN=MIDDLE  ALT=\"ln%2850%29+=+ln%282000%29+-+.03t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "subtract ln(2000) from both sides of this equation and then divide both sides of this equation by (-.03) to get:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%2850%29+-+ln%282000%29%29%2F%28-.03%29+=+t ALIGN=MIDDLE  ALT=\"%28ln%2850%29+-+ln%282000%29%29%2F%28-.03%29+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283.912023005+-+7.60090246%29%2F%28-.03%29+=+t ALIGN=MIDDLE  ALT=\"%283.912023005+-+7.60090246%29%2F%28-.03%29+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=122.9626485+=+t ALIGN=MIDDLE  ALT=\"122.9626485+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The population will decay to 50 in 122.9626485 hours.<BR>\n" );
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document.write( "substitute this value for t in the original equation to get:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28t%29=2000%2Ae%5E%28-.03%2A%28122.9626485%29%29 ALIGN=MIDDLE  ALT=\"B%28t%29=2000%2Ae%5E%28-.03%2A%28122.9626485%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28t%29+=+50 ALIGN=MIDDLE  ALT=\"B%28t%29+=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" > confirming that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+122.9626485 ALIGN=MIDDLE  ALT=\"t+=+122.9626485\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a good value for t.<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );