document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1108963>Question 1108963</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The count in a bateria culture was 400 after 15 minutes and 2000 after 35 minutes. Assuming the count grows exponentially,\r<BR>\n" );
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document.write( "What was the initial size of the culture? <BR>\n" );
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document.write( "Find the doubling period. <BR>\n" );
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document.write( "Find the population after 110 minutes. <BR>\n" );
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document.write( "When will the population reach 10000. <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 #724004 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=724004\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ab%5Ex ALIGN=MIDDLE  ALT=\"y+=+ab%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000+=+ab%5E35 ALIGN=MIDDLE  ALT=\"2000+=+ab%5E35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400+=+ab%5E15 ALIGN=MIDDLE  ALT=\"400+=+ab%5E15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Divide the two equations, eliminating a:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+b%5E20 ALIGN=MIDDLE  ALT=\"5+=+b%5E20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+5%5E%281%2F20%29+=+1.083798387 ALIGN=MIDDLE  ALT=\"b+=+5%5E%281%2F20%29+=+1.083798387\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Keep at least 4 or 5 decimal places if you want your answers to the later parts of the problem to be accurate.<br><BR>\n" );
document.write( "Question 1: initial population.<br><BR>\n" );
document.write( "The initial population is the population after 15 minutes, divided by b^15:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+400%2Fb%5E15+=+119.6279 ALIGN=MIDDLE  ALT=\"y+=+400%2Fb%5E15+=+119.6279\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This of course is a nonsensical answer; bacteria are counted in whole numbers.  However, again you need to keep several decimal places in order for your answers to be accurate.<br><BR>\n" );
document.write( "Answer 1: The initial population was a = 119.6279.<br><BR>\n" );
document.write( "Question 2: doubling time<br><BR>\n" );
document.write( "b is the growth factor each minute; you want to know how many minutes it takes for the growth to be double:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ex+=+2 ALIGN=MIDDLE  ALT=\"b%5Ex+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%28b%29+=+log%282%29 ALIGN=MIDDLE  ALT=\"x%2Alog%28b%29+=+log%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%282%29%2Flog%28b%29+=+8.36153 ALIGN=MIDDLE  ALT=\"x+=+log%282%29%2Flog%28b%29+=+8.36153\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Answer 2: the doubling time is 8.36153 minutes.<br><BR>\n" );
document.write( "Question 3: the population after 110 minutes<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%5E110+=+835925 ALIGN=MIDDLE  ALT=\"ab%5E110+=+835925\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Answer 3: the population after 110 minutes is 835925.<br><BR>\n" );
document.write( "Question 4: the time when the population reaches 10000.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%5Ex+=+10000 ALIGN=MIDDLE  ALT=\"ab%5Ex+=+10000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ex+=+10000%2Fa ALIGN=MIDDLE  ALT=\"b%5Ex+=+10000%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%28b%29+=+log%2810000%2Fa%29+=+4-log%28a%29 ALIGN=MIDDLE  ALT=\"x%2Alog%28b%29+=+log%2810000%2Fa%29+=+4-log%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284-log%28a%29%29%2Flog%28b%29+=+55 ALIGN=MIDDLE  ALT=\"x+=+%284-log%28a%29%29%2Flog%28b%29+=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Answer 4: the population reaches 10000 at 55 minutes.<br><BR>\n" );
document.write( "Note we could have answered question 4 without doing any difficult calculations.  The population increased by a factor of 5, from 400 to 2000, in 20 minutes (between 15 minutes and 35 minutes).  10000 is 5 times 2000, so a population of 10000 will be reached 20 minutes after it was 2000; 20 minutes after 35 minutes is 55 minutes.<br><BR>\n" );
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