document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=885241>Question 885241</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The doubling period of a baterial population is 15 minutes. At time t = 90 minutes, the baterial population was 50000. Round your answers to at least 1 decimal place.\r<BR>\n" );
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document.write( "What was the initial population at time t = 0 ?  \r<BR>\n" );
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document.write( "Find the size of the baterial population after 4 hours </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535041 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=535041\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> WITHOUT FORMULAS OR CALCULATOR (but with understanding):<BR>\n" );
document.write( "The population of that kind of bacteria doubles every 15 minutes, and after 90 minutes there's 50,000 bacteria.<BR>\n" );
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document.write( "At time t=90 minutes, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%2F15=6 ALIGN=MIDDLE  ALT=\"90%2F15=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > doublings have happened since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The population at t=90 minutes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6=64 ALIGN=MIDDLE  ALT=\"2%5E6=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > times the population at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the population at t=0 must have been<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50000%2F2%5E6=1541.25 ALIGN=MIDDLE  ALT=\"50000%2F2%5E6=1541.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I do not need to round that number, because it is an exact calculation.<BR>\n" );
document.write( "Saying that at time t=0 there were <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1541%261%2F4 ALIGN=MIDDLE  ALT=\"1541%261%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > bacteria is a little silly.<BR>\n" );
document.write( "I expect the number of bacteria to be an integer.<BR>\n" );
document.write( "However, the problem asks to round to at least 1 decimal place, so let's be silly.<BR>\n" );
document.write( "How did I calculate that result? With pencil and paper.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50000%2F2%5E6=5%2A10%5E4%2F2%5E6=5%2A%285%2A2%29%5E4%2F2%5E6=5%2A5%5E4%2A2%5E4%2F2%5E6=565%2F2%5E2=3125%2F4=781.25 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "At t=4 hours, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%2860min%2F%2215+min%22%29=4%2A4=16 ALIGN=MIDDLE  ALT=\"4%2A%2860min%2F%2215+min%22%29=4%2A4=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > doubling period have passed since t=0.<BR>\n" );
document.write( "That is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-6=10 ALIGN=MIDDLE  ALT=\"16-6=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > doubling periods after t=90 minutes,<BR>\n" );
document.write( "so the number of bacteria should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E10 ALIGN=MIDDLE  ALT=\"2%5E10\"   DISABLED_style= \"max-width:90%; height:auto;\" > times what it was at 90 minutes.<BR>\n" );
document.write( "That is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50000%2A2%5E10=%2251%2C200%2C000%22 ALIGN=MIDDLE  ALT=\"50000%2A2%5E10=%2251%2C200%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that is an exact number too.<BR>\n" );
document.write( "Should I say it is 51,200,000.00 to have at least one decimal place expressly stated.<BR>\n" );
document.write( "How did I calculate that result? With pencil and paper.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50000%2A2%5E10=%285%2A10%5E4%29%2A2%5E10=5%2A10%5E4%2A2%2A2%5E9=%285%2A2%29%2A10%5E4%2A2%5E9=10%2A10%5E4%2A2%5E9=10%5E5%2A2%5E9=10%5E5%2A512=51.2%2A10%5E6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH FORMULAS AND CALCULATOR:<BR>\n" );
document.write( "We are having a case of exponential growth.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29 ALIGN=MIDDLE  ALT=\"P%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= population at t=t minutes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%280%29%2A2%5E%28%22t+%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%28t%29=P%280%29%2A2%5E%28%22t+%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29%2FP%280%29=2%5E%28%22t+%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%28t%29%2FP%280%29=2%5E%28%22t+%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since calculators allow us to calculate natural logarithms, and base 10 logarithms,<BR>\n" );
document.write( "we can also write that as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28P%28t%29%2FP%280%29%29=%28t%2F15%29%2Aln%282%29 ALIGN=MIDDLE  ALT=\"ln%28P%28t%29%2FP%280%29%29=%28t%2F15%29%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+log%28P%28t%29%2FP%280%29%29=%28t%2F15%29%2Alog%282%29 ALIGN=MIDDLE  ALT=\"+log%28P%28t%29%2FP%280%29%29=%28t%2F15%29%2Alog%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Going from the natural logarithms version to exponentials on base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29%2FP%280%29=e%5E%28%22ln%282%29t+%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%28t%29%2FP%280%29=e%5E%28%22ln%282%29t+%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%280%29%2Ae%5E%28%22ln%282%29t+%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%28t%29=P%280%29%2Ae%5E%28%22ln%282%29t+%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Those are very popular forms. The approximation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29=0.693 ALIGN=MIDDLE  ALT=\"ln%282%29=0.693\"   DISABLED_style= \"max-width:90%; height:auto;\" > is often used.<BR>\n" );
document.write( " <BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=90 ALIGN=MIDDLE  ALT=\"t=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2890%29%2FP%280%29=e%5E%28%22ln%282%2990%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%2890%29%2FP%280%29=e%5E%28%22ln%282%2990%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50000%2FP%280%29=e%5E%286ln%282%29%29 ALIGN=MIDDLE  ALT=\"50000%2FP%280%29=e%5E%286ln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29%2F50000=e%5E%28-6ln%282%29%29 ALIGN=MIDDLE  ALT=\"P%280%29%2F50000=e%5E%28-6ln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=50000%2Ae%5E%28-+6ln%282%29%29 ALIGN=MIDDLE  ALT=\"P%280%29=50000%2Ae%5E%28-+6ln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=50000%2Ae%5E%28-+6ln%282%29%29 ALIGN=MIDDLE  ALT=\"P%280%29=50000%2Ae%5E%28-+6ln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=781.25 ALIGN=MIDDLE  ALT=\"P%280%29=781.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "}}} (using the calculator's value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29 ALIGN=MIDDLE  ALT=\"ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "(Using 0.693 will give you a different decimal part).<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4hours=240minutes ALIGN=MIDDLE  ALT=\"t=4hours=240minutes\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28240%29%2FP%280%29=e%5E%28%22ln%282%29240%2F+15%22%29 ALIGN=MIDDLE  ALT=\"P%28240%29%2FP%280%29=e%5E%28%22ln%282%29240%2F+15%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28240%29%2F781.25=e%5E%2816%2Aln%282%29%29 ALIGN=MIDDLE  ALT=\"P%28240%29%2F781.25=e%5E%2816%2Aln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28240%29=781.25%2Ae%5E%2816%2Aln%282%29%29 ALIGN=MIDDLE  ALT=\"P%28240%29=781.25%2Ae%5E%2816%2Aln%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28240%29=%2251%2C200%2C000%22 ALIGN=MIDDLE  ALT=\"P%28240%29=%2251%2C200%2C000%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );