document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=998640>Question 998640</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help. Show step by step. Thanks \r<BR>\n" );
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document.write( "A bacteria culture starts with 500  bacteria and grows at a rate proportional to its size. After 3  hours there will be 1500  bacteria.\r<BR>\n" );
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document.write( "(a) Express the population P  after t  hours as a function of t . <BR>\n" );
document.write( "P(t) =        \r<BR>\n" );
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document.write( " (b) What will be the population after 5 hours? <BR>\n" );
document.write( "      bacteria\r<BR>\n" );
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document.write( " (c) How long will it take for the population to reach 2130? Give your answer accurate to at least 2 decimal places.<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 #616412 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=616412\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> That is what we cal exponential growth.<BR>\n" );
document.write( "We like to express it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%5B0%5D%2Ae%5Ekt ALIGN=MIDDLE  ALT=\"P%28t%29=P%5B0%5D%2Ae%5Ekt\"   DISABLED_style= \"max-width:90%; height:auto;\" > based on<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5B0%5D ALIGN=MIDDLE  ALT=\"P%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the initial population,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >= a rate constant that we will find, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" >= and irrational number.<BR>\n" );
document.write( "We could use any other base, as in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%5B0%5D%2A10%5EKt ALIGN=MIDDLE  ALT=\"P%28t%29=P%5B0%5D%2A10%5EKt\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with a power of base <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a different constant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K ALIGN=MIDDLE  ALT=\"K\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, we really like to use the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a base,<BR>\n" );
document.write( "because it makes the growth rate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dP%2Fdt=k%2AP%28t%29 ALIGN=MIDDLE  ALT=\"dP%2Fdt=k%2AP%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "making <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 ratio of the growth rate to the population size.\r<BR>\n" );
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document.write( "(a) So, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5B0%5D=500 ALIGN=MIDDLE  ALT=\"P%5B0%5D=500\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%283%29=1500 ALIGN=MIDDLE  ALT=\"P%283%29=1500\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Ae%5E%283k%29=1500 ALIGN=MIDDLE  ALT=\"500%2Ae%5E%283k%29=1500\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%283k%29=1500%2F500 ALIGN=MIDDLE  ALT=\"e%5E%283k%29=1500%2F500\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%283k%29=3 ALIGN=MIDDLE  ALT=\"e%5E%283k%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3k=ln%283%29 ALIGN=MIDDLE  ALT=\"3k=ln%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=ln%283%29%2F3 ALIGN=MIDDLE  ALT=\"k=ln%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28t%29=500%2Ae%5E%28%22ln%283%29t%2F3%22%29%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28t%29=500%2Ae%5E%28%22ln%283%29t%2F3%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28t%29=500%2A3%5E%28%22t%2F3%22%29%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28t%29=500%2A3%5E%28%22t%2F3%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28t%29=500%2Aroot%283%2C3%5Et%29%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28t%29=500%2Aroot%283%2C3%5Et%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%285%29=500%2Ae%5E%28%225ln%283%29%2F3%22%29=highlight%283120%29 ALIGN=MIDDLE  ALT=\"P%285%29=500%2Ae%5E%28%225ln%283%29%2F3%22%29=highlight%283120%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded)<BR>\n" );
document.write( "(c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500%2Ae%5E%28%22ln%283%29t%2F3%22%29=2130 ALIGN=MIDDLE  ALT=\"500%2Ae%5E%28%22ln%283%29t%2F3%22%29=2130\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28%22ln%283%29t%2F3%22%29=2130%2F500 ALIGN=MIDDLE  ALT=\"e%5E%28%22ln%283%29t%2F3%22%29=2130%2F500\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28%22ln%283%29t%2F3%22%29=4.26 ALIGN=MIDDLE  ALT=\"e%5E%28%22ln%283%29t%2F3%22%29=4.26\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29t%2F3=ln%284.26%29 ALIGN=MIDDLE  ALT=\"ln%283%29t%2F3=ln%284.26%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2F3=ln%284.26%29%2Fln%283%29 ALIGN=MIDDLE  ALT=\"t%2F3=ln%284.26%29%2Fln%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3ln%284.26%29%2Fln%283%29 ALIGN=MIDDLE  ALT=\"t=3ln%284.26%29%2Fln%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28t=3.96%29 ALIGN=MIDDLE  ALT=\"highlight%28t=3.96%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >(rounded) </SPAN>\n" );
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