document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=993293>Question 993293</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The doubling period of a bacterial population is 20  minutes. At time t=90  minutes, the bacterial population was 50000.\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 bacterial population after 3 hours<BR>\n" );
document.write( "*formula: p(t)=a(b)^t<BR>\n" );
document.write( "**Note:b=1+r<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 #612585 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39623)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=612585\">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=a%2Ab%5Et ALIGN=MIDDLE  ALT=\"p=a%2Ab%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you have some information for doubling time.  Twenty minutes is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > hour.  Try to use just that much and see what you can find.\r<BR>\n" );
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document.write( "Try taking a as the initial population when t=0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28t=1%2F3%2Cp=2%2Ca=1%29 ALIGN=MIDDLE  ALT=\"system%28t=1%2F3%2Cp=2%2Ca=1%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=2=1%2Ab%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"2=1%2Ab%5E%281%2F3%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=b=root%283%2C2%29 ALIGN=MIDDLE  ALT=\"b=root%283%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or b=1.26 approximately.<BR>\n" );
document.write( "The model might work as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=a%2A%281.26%29%5Et ALIGN=MIDDLE  ALT=\"p=a%2A%281.26%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The next part of the description is the given point  (1.5, 50000), and you want to know a, or value of p when t=0 instead of t=1.5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%281.26%29%5E%283%2F2%29=50000 ALIGN=MIDDLE  ALT=\"a%2A%281.26%29%5E%283%2F2%29=50000\"   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=a=50000%2F%281.26%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"a=50000%2F%281.26%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and for convenience, recall where the \"1.26\" came from:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=50000%2F%28root%283%2C2%29%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"a=50000%2F%28root%283%2C2%29%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "...the rendering does not look properly aligned but that is an equation, formula for a =  fifty thousand over (cube root of two) to the three-halves power...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=50000%2F%282%5E%281%2F3%29%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"a=50000%2F%282%5E%281%2F3%29%29%5E%283%2F2%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=a=50000%2F%282%5E%28%281%2F3%29%283%2F2%29%29%29 ALIGN=MIDDLE  ALT=\"a=50000%2F%282%5E%28%281%2F3%29%283%2F2%29%29%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=a=50000%2F%282%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"a=50000%2F%282%5E%281%2F2%29%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=highlight%28a=50000%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"highlight%28a=50000%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simple radical form, although denominator is not rationalized.\r<BR>\n" );
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document.write( "This would be about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=35400%29 ALIGN=MIDDLE  ALT=\"highlight%28a=35400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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