document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=757191>Question 757191</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question is based on grad 12, Calculus<BR>\n" );
document.write( "a bacteria colony has a mass that varies with time.The original mass of the bacteria is 20 g and the mass increases by 50% eache day\r<BR>\n" );
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document.write( "a) write an equation to model the growth of the mass of the colony <BR>\n" );
document.write( "b) what is the intial rate of change of the mass of the colony?<BR>\n" );
document.write( "c) During which day is the mass of the colony increasing by 60 g/day?<BR>\n" );
document.write( "d) When will the mass of the colony be 60 Kg? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #460702 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=460702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The mass of the colony increases by a factor of 1.5 each day,<BR>\n" );
document.write( "so if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time elapsed, in days,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >= mass of bacteria, in grams,<BR>\n" );
document.write( "we could write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28t%29=20%2A1.5%5Et ALIGN=MIDDLE  ALT=\"m%28t%29=20%2A1.5%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "However, it is traditional (and more convenient when calculating derivatives) to use<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28m%28t%29=20%2Ae%5E%28t%2Aln%281.5%29%29%29 ALIGN=MIDDLE  ALT=\"highlight%28m%28t%29=20%2Ae%5E%28t%2Aln%281.5%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <-- answer to part a) of the problem<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) The rate of change of the mass of the colony would be the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28t%29 ALIGN=MIDDLE  ALT=\"m%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dm%2Fdt=20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29 ALIGN=MIDDLE  ALT=\"dm%2Fdt=20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Fot <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that rate is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2Aln%281.5%29 ALIGN=MIDDLE  ALT=\"20%2Aln%281.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= approximately 8.1 g/day<BR>\n" );
document.write( " <BR>\n" );
document.write( "c) During which day is the mass of the colony increasing by 60 g/day?<BR>\n" );
document.write( "we need to find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > that makes the rate, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dm%2Fdt=20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29 ALIGN=MIDDLE  ALT=\"dm%2Fdt=20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equal to 60 g/day<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29=60 ALIGN=MIDDLE  ALT=\"20%2Aln%281.5%29%2Ae%5E%28t%2Aln%281.5%29%29=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28t%2Aln%281.5%29%29=60%2F20%2Fln%281.5%29 ALIGN=MIDDLE  ALT=\"e%5E%28t%2Aln%281.5%29%29=60%2F20%2Fln%281.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2Aln%281.5%29=ln%283%2Fln%281.5%29%29 ALIGN=MIDDLE  ALT=\"t%2Aln%281.5%29=ln%283%2Fln%281.5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=ln%283%2Fln%281.5%29%29%2Fln%281.5%29 ALIGN=MIDDLE  ALT=\"t=ln%283%2Fln%281.5%29%29%2Fln%281.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> t = approx. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.94 ALIGN=MIDDLE  ALT=\"4.94\"   DISABLED_style= \"max-width:90%; height:auto;\" > days.<BR>\n" );
document.write( "I would call that time (and any time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%3C=t%3C5 ALIGN=MIDDLE  ALT=\"4%3C=t%3C5\"   DISABLED_style= \"max-width:90%; height:auto;\" > between 4 and 5) day <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285%29 ALIGN=MIDDLE  ALT=\"highlight%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "d) When will the mass of the colony be 60 Kg?<BR>\n" );
document.write( "60Kg is a lot. It's 60,000 grams.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28t%29=20%2Ae%5E%28t%2Aln%281.5%29%29=60000 ALIGN=MIDDLE  ALT=\"m%28t%29=20%2Ae%5E%28t%2Aln%281.5%29%29=60000\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28t%2Aln%281.5%29%29=60000%2F20 ALIGN=MIDDLE  ALT=\"e%5E%28t%2Aln%281.5%29%29=60000%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2Aln%281.5%29=ln%283000%29 ALIGN=MIDDLE  ALT=\"t%2Aln%281.5%29=ln%283000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=ln%283000%29%2Fln%281.5%29 ALIGN=MIDDLE  ALT=\"t=ln%283000%29%2Fln%281.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=about19.75 ALIGN=MIDDLE  ALT=\"t=about19.75\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we approximate to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=highlight%2820days%29 ALIGN=MIDDLE  ALT=\"t=highlight%2820days%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, in the ideal world of math, it would take 20 days to get 60 kg of bacteria.<BR>\n" );
document.write( " <BR>\n" );
document.write( "In real life, a colony would not keep growing at the same rate for that long. Even with a lot of attention, \"feeding\", and \"transplanting\", grow would not stay constant. </SPAN>\n" );
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