document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157354>Question 1157354</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The size of certain insect poplation is given by <BR>\n" );
document.write( " P(t)  =2000 e^(0.01t), where t is measured in days. <BR>\n" );
document.write( " a) How many insects were present initially? <BR>\n" );
document.write( " b) what is the growth rate? <BR>\n" );
document.write( " c) At what time will the population double? <BR>\n" );
document.write( " d) give a differential equation satisfied by p(t)? <BR>\n" );
document.write( " d) how fast the becteria is growing when it reaches 10.000? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #780258 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=780258\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The size of certain insect population is given by\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=2000+e%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=2000+e%5E%280.01t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is measured in days.\r<BR>\n" );
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document.write( "a) How many insects were present initially?\r<BR>\n" );
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document.write( " the initial amount is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=2000+ ALIGN=MIDDLE  ALT=\"P%280%29=2000+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b) what is the growth rate?\r<BR>\n" );
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document.write( "The growth rate is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.01 ALIGN=MIDDLE  ALT=\"0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "c) At what time will the population double?\r<BR>\n" );
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document.write( " double amount is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=4000 ALIGN=MIDDLE  ALT=\"P%28t%29+=4000\"   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=4000+=2000+e%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"4000+=2000+e%5E%280.01t%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=4000%2F2000+=e%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"4000%2F2000+=e%5E%280.01t%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=e%5E%280.01t%29=2 ALIGN=MIDDLE  ALT=\"e%5E%280.01t%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >......take natural log of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%280.01t%29%29=ln%282%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%280.01t%29%29=ln%282%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=%280.01t%29ln%28e%29=ln%282%29 ALIGN=MIDDLE  ALT=\"%280.01t%29ln%28e%29=ln%282%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=0.01t=ln%282%29%2Fln%28e%29 ALIGN=MIDDLE  ALT=\"0.01t=ln%282%29%2Fln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%29=1 ALIGN=MIDDLE  ALT=\"ln%28e%29=1\"   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=0.01t=0.69314718 ALIGN=MIDDLE  ALT=\"0.01t=0.69314718\"   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=t=0.69314718%2F0.01 ALIGN=MIDDLE  ALT=\"t=0.69314718%2F0.01\"   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=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >≈<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=69.315+ ALIGN=MIDDLE  ALT=\"69.315+\"   DISABLED_style= \"max-width:90%; height:auto;\" >days\r<BR>\n" );
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document.write( "d) give a differential equation satisfied by <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;\" >?\r<BR>\n" );
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document.write( "take the constant out : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2A+f+%29 ALIGN=MIDDLE  ALT=\"%28a%2A+f+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >* <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   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=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%29+=2000+%2A%28d%2Fdt%29%28e%5E%280.01t%29%29 ALIGN=MIDDLE  ALT=\"%28t%29+=2000+%2A%28d%2Fdt%29%28e%5E%280.01t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "apply the chain rule:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28df%28u%29%2Fdt%29=%28df%2Fdu%29%2A%28du%2Fdx%29 ALIGN=MIDDLE  ALT=\"%28df%28u%29%2Fdt%29=%28df%2Fdu%29%2A%28du%2Fdx%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=f=e%5Eu ALIGN=MIDDLE  ALT=\"f=e%5Eu\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=0.01t ALIGN=MIDDLE  ALT=\"u=0.01t\"   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=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%29+=2000+%2A%28d%2Fdu%29%28e%5Eu%29%2A%28d%2Fdt%29%280.01t%29 ALIGN=MIDDLE  ALT=\"%28t%29+=2000+%2A%28d%2Fdu%29%28e%5Eu%29%2A%28d%2Fdt%29%280.01t%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=%28d%2Fdu%29%28e%5Eu%29=e%5Eu ALIGN=MIDDLE  ALT=\"%28d%2Fdu%29%28e%5Eu%29=e%5Eu\"   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=%28d%2Fdt%29%280.01t%29=0.01 ALIGN=MIDDLE  ALT=\"%28d%2Fdt%29%280.01t%29=0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > back\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%29=2000%2Ae%5E%280.01t%29%2A0.01 ALIGN=MIDDLE  ALT=\"%28t%29=2000%2Ae%5E%280.01t%29%2A0.01\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >'<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%29+=+20%2Ae%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"%28t%29+=+20%2Ae%5E%280.01t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "d) how fast the bacteria is growing when it reaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   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=10000+=2000%2Ae%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"10000+=2000%2Ae%5E%280.01t%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=10000+%2F2000+=e%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"10000+%2F2000+=e%5E%280.01t%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=5+=e%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"5+=e%5E%280.01t%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=ln%28e%5E%280.01t%29%29=ln%285%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%280.01t%29%29=ln%285%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=%280.01t%29ln%28e%29=ln%285%29 ALIGN=MIDDLE  ALT=\"%280.01t%29ln%28e%29=ln%285%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=0.01t=ln%285%29 ALIGN=MIDDLE  ALT=\"0.01t=ln%285%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=t=ln%285%29%2F0.01 ALIGN=MIDDLE  ALT=\"t=ln%285%29%2F0.01\"   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=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >≈<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=160.944+ ALIGN=MIDDLE  ALT=\"160.944+\"   DISABLED_style= \"max-width:90%; height:auto;\" >days\r<BR>\n" );
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document.write( " </SPAN>\n" );
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