document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87056>Question 87056</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The logistic growth model P(t)=1270/1+27.22e^-0.348t represents the population of a bacterium in a culture tube after \"t\" hours.<BR>\n" );
document.write( "A) What was the initial amount of bacteria in the population?<BR>\n" );
document.write( "B)After how many hours is the population of bacteria 1000? Round to the nearest hour.<BR>\n" );
document.write( "C) What is the limitig size of P(t), the poluation of bacterium? \r<BR>\n" );
document.write( "\n" );
document.write( "* For B) I put 1000 in for \"P\" and C) I think the answer would be 1270...right? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63025 by </B><A HREF=/tutors/aboutme.mpl?userid=Nate><B>Nate(3500)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Nate><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=63025\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A) What was the initial amount of bacteria in the population?<BR>\n" );
document.write( "Initial amount would be determined after 0 hours ... t = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%29%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%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=P%280%29+=+1270%2F%281+%2B+27.22e%5E%280%29%29 ALIGN=MIDDLE  ALT=\"P%280%29+=+1270%2F%281+%2B+27.22e%5E%280%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=P%280%29+=+1270%2F%281+%2B+27.22%29 ALIGN=MIDDLE  ALT=\"P%280%29+=+1270%2F%281+%2B+27.22%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%280%29+=+1270%2F28.22 ALIGN=MIDDLE  ALT=\"P%280%29+=+1270%2F28.22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Initial amount would be about 45<BR>\n" );
document.write( "B)After how many hours is the population of bacteria 1000? Round to the nearest hour.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%29%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%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=1000+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%29%29 ALIGN=MIDDLE  ALT=\"1000+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%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=1+%2B+27.22e%5E%28-0.348t%29+=+1270%2F1000 ALIGN=MIDDLE  ALT=\"1+%2B+27.22e%5E%28-0.348t%29+=+1270%2F1000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+27.22e%5E%28-0.348t%29+=+1.27 ALIGN=MIDDLE  ALT=\"1+%2B+27.22e%5E%28-0.348t%29+=+1.27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27.22e%5E%28-0.348t%29+=+0.27 ALIGN=MIDDLE  ALT=\"27.22e%5E%28-0.348t%29+=+0.27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-0.348t%29+=+0.27%2F27.22 ALIGN=MIDDLE  ALT=\"e%5E%28-0.348t%29+=+0.27%2F27.22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2Ce%5E%28-0.348t%29%29+=+log%28e%2C0.27%2F27.22%29 ALIGN=MIDDLE  ALT=\"log%28e%2Ce%5E%28-0.348t%29%29+=+log%28e%2C0.27%2F27.22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.348t+=+log%28e%2C0.27%2F27.22%29 ALIGN=MIDDLE  ALT=\"-0.348t+=+log%28e%2C0.27%2F27.22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-log%28e%2C0.27%2F27.22%29%2F0.348 ALIGN=MIDDLE  ALT=\"t+=+-log%28e%2C0.27%2F27.22%29%2F0.348\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "C) What is the limitig size of P(t), the poluation of bacterium?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%29%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+1270%2F%281+%2B+27.22e%5E%28-0.348t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Firstly, we would have to map out 1 + 27.22e^(-0.348t). Since time is positive, we must see how 1 + 27.22e^(-0.348t) reacts as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases positively.<BR>\n" );
document.write( "Red: 1 + 27.22/e^(0.348t)<BR>\n" );
document.write( "Green: 1270/(1 + 27.22e^(-0.348t))<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C800%2C-20%2C50%2C-2%2C1400%2C1+%2B+27.22%2F2.718%5E%280.348x%29%2C1270%2F%281+%2B+27.22%2F%282.718%29%5E%280.348x%29%29%2C1270%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "1270 is being divided by an exponentially smaller number every hour<BR>\n" );
document.write( "as 1 + 27.22/e^(0.348t) approaches infinity for time ... it equals 1<BR>\n" );
document.write( "1270 / 1 is the limit </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );