document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=885246>Question 885246</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The count in a bateria culture was 600 after 20 minutes and 1600 after 30 minutes.\r<BR>\n" );
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document.write( "What was the initial size of the culture?  \r<BR>\n" );
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document.write( "Find the doubling period.  \r<BR>\n" );
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document.write( "Find the population after 90 minutes.  \r<BR>\n" );
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document.write( "When will the population reach 11000 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535036 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39618)</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=535036\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You may be able to do this using hours instead of minutes.\r<BR>\n" );
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document.write( "I = initial count of bacteria<BR>\n" );
document.write( "p = count of bacteria after time t<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28p=Ie%5E%28kt%29%29 ALIGN=MIDDLE  ALT=\"highlight_green%28p=Ie%5E%28kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, exponential growth formula\r<BR>\n" );
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document.write( "Interesting that we do not know the count of bacteria at time t=0.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28p%29=ln%28Ie%5E%28kt%29%29 ALIGN=MIDDLE  ALT=\"ln%28p%29=ln%28Ie%5E%28kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28p%29=ln%28I%29%2Bln%28e%5E%28kt%29%29 ALIGN=MIDDLE  ALT=\"ln%28p%29=ln%28I%29%2Bln%28e%5E%28kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28p%29=ln%28I%29%2Bkt%2A1 ALIGN=MIDDLE  ALT=\"ln%28p%29=ln%28I%29%2Bkt%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28ln%28p%29=ln%28I%29%2Bkt%29 ALIGN=MIDDLE  ALT=\"highlight_green%28ln%28p%29=ln%28I%29%2Bkt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a LINEAR equation.  The vertical axis intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28I%29 ALIGN=MIDDLE  ALT=\"ln%28I%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The vertical axis variable is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28p%29 ALIGN=MIDDLE  ALT=\"ln%28p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and is a function of time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "You have two data points to help in getting vertical intercept and slope.<BR>\n" );
document.write( "20 & 600;<BR>\n" );
document.write( "30 & 1600.<BR>\n" );
document.write( "The way you must use them first is this way:<BR>\n" );
document.write( "20 & ln(600)=6.3969<BR>\n" );
document.write( "30 & ln(1600)=7.3778<BR>\n" );
document.write( "Meaning the two points for the LINEAR form of this description are these:<BR>\n" );
document.write( "(20, 6.3969)  and  (30, 7.3778).<BR>\n" );
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document.write( "k or SLOPE<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%287.3778-6.3969%29%2F%2830-20%29 ALIGN=MIDDLE  ALT=\"k=%287.3778-6.3969%29%2F%2830-20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0.9809%2F10 ALIGN=MIDDLE  ALT=\"k=0.9809%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0.09809 ALIGN=MIDDLE  ALT=\"k=0.09809\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Vertical Intercept<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28p%29=ln%28I%29%2Bkt ALIGN=MIDDLE  ALT=\"ln%28p%29=ln%28I%29%2Bkt\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28I%29=ln%28p%29-kt ALIGN=MIDDLE  ALT=\"ln%28I%29=ln%28p%29-kt\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Pick either point to finish finding ln(I).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28I%29=7.3778-0.09809%2A30 ALIGN=MIDDLE  ALT=\"ln%28I%29=7.3778-0.09809%2A30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28I%29=4.4351 ALIGN=MIDDLE  ALT=\"ln%28I%29=4.4351\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Antilog, base e, for 4.4351 will be I.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I=84 ALIGN=MIDDLE  ALT=\"I=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >,   more sensible than saying 84.36...<BR>\n" );
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document.write( "Build back the exponential growth equation using the now found values:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28p=84%2Ae%5E%280.09809t%29%29 ALIGN=MIDDLE  ALT=\"highlight%28p=84%2Ae%5E%280.09809t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You can answer whatever questions you want for this fictional or factual bacteria culture.\r<BR>\n" );
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document.write( "<i>Find the doubling period.  </i><BR>\n" );
document.write( "Just let I=1 and p=2.  Now find t.\r<BR>\n" );
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document.write( "<i>Find the population after 90 minutes.  </i><BR>\n" );
document.write( "Obviously, just let t=90...\r<BR>\n" );
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document.write( "<i>When will the population reach 11000</i><BR>\n" );
document.write( "Let p=11000 and solve for t.  There are clues how to do these i some of the steps I showed. </SPAN>\n" );
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