document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=214807>Question 214807</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 growing exponentially according to the function below, where t is the time in hours. How long will it take for the population of the colony to grow to 1,000?\r<BR>\n" );
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document.write( "B(t)=12*e^(0.2*t)\r<BR>\n" );
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document.write( "e=2.718 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #162267 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=162267\">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=12%2Ae%5E%28%280.2%2At%29%29 ALIGN=MIDDLE  ALT=\"B%28t%29=12%2Ae%5E%28%280.2%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "substitute 1000 for B(t) to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000+=+12%2Ae%5E%28%28.2%2At%29%29 ALIGN=MIDDLE  ALT=\"1000+=+12%2Ae%5E%28%28.2%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><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%281000%29+=+ln%2812%2Ae%5E%28.2%2At%29%29 ALIGN=MIDDLE  ALT=\"ln%281000%29+=+ln%2812%2Ae%5E%28.2%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "couple of general properties of logarithms are applicable.<BR>\n" );
document.write( "the first is:<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( "the second is:<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( "using the first one, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2812%2Ae%5E%28.2%2At%29%29+=+ln%2812%29+%2B+ln%28e%5E%28%28.2%2At%29%29%29 ALIGN=MIDDLE  ALT=\"ln%2812%2Ae%5E%28.2%2At%29%29+=+ln%2812%29+%2B+ln%28e%5E%28%28.2%2At%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "using the second one:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%28.2%2At%29%29+=+.2%2At%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%28.2%2At%29%29+=+.2%2At%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%281000%29+=+ln%2812%29+%2B+.2%2At%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%281000%29+=+ln%2812%29+%2B+.2%2At%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2812%29 ALIGN=MIDDLE  ALT=\"ln%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides of this equation to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%281000%29+-+ln%2812%29++=+.2%2At%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%281000%29+-+ln%2812%29++=+.2%2At%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "divide both sides of this equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.2%2Aln%28e%29 ALIGN=MIDDLE  ALT=\".2%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%281000%29+-+ln%2812%29%29+%2F+%28ln%28e%29%2A.2%29+=+t ALIGN=MIDDLE  ALT=\"%28ln%281000%29+-+ln%2812%29%29+%2F+%28ln%28e%29%2A.2%29+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286.907755279+-+2.48490665%29%2F%281%2A%28.2%29%29 ALIGN=MIDDLE  ALT=\"%286.907755279+-+2.48490665%29%2F%281%2A%28.2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "solve for t to get:<BR>\n" );
document.write( "t = 22.11424315 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the population should increase to 1000 in 22.11424315 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "substitute in original equation of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28t%29=12%2Ae%5E%28%280.2%2At%29%29 ALIGN=MIDDLE  ALT=\"B%28t%29=12%2Ae%5E%28%280.2%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28t%29=12%2Ae%5E%28%280.2%2A%2822.11424315%29%29%29 ALIGN=MIDDLE  ALT=\"B%28t%29=12%2Ae%5E%28%280.2%2A%2822.11424315%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:<BR>\n" );
document.write( "B(t) = 1000 confirming t = 22.11424315 hours is a good value for t.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Colony will increase to 1000 in 22.11424315 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );