document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=135705>Question 135705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The count in a bacterial culture was 400 after 2 hours and 25,600 after 6 hours.<BR>\n" );
document.write( "     We are assuming exponential growth.<BR>\n" );
document.write( "	<BR>\n" );
document.write( "	What is the rate of growth of the population of bacteria?<BR>\n" );
document.write( "	What was the initial population at time t = 0 hours?<BR>\n" );
document.write( "	Write the function that models the population n(t) after t hours.<BR>\n" );
document.write( "	When will the number of bacteria exceed 100,000? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #99449 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=99449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%2Ab%5Et ALIGN=MIDDLE  ALT=\"y=a%2Ab%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general exponential equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400=a%2Ab%5E2 ALIGN=MIDDLE  ALT=\"400=a%2Ab%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2 ALIGN=MIDDLE  ALT=\"t=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=400 ALIGN=MIDDLE  ALT=\"y=400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400%2Fb%5E2=a ALIGN=MIDDLE  ALT=\"400%2Fb%5E2=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate \"a\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%2Ab%5Et ALIGN=MIDDLE  ALT=\"y=a%2Ab%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general exponential equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25600=a%2Ab%5E6 ALIGN=MIDDLE  ALT=\"25600=a%2Ab%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=6 ALIGN=MIDDLE  ALT=\"t=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=25600 ALIGN=MIDDLE  ALT=\"y=25600\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25600=%28400%2Fb%5E2%29%2Ab%5E6 ALIGN=MIDDLE  ALT=\"25600=%28400%2Fb%5E2%29%2Ab%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=6 ALIGN=MIDDLE  ALT=\"t=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=400%2Fb%5E2 ALIGN=MIDDLE  ALT=\"a=400%2Fb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25600=400b%5E6%2Fb%5E2 ALIGN=MIDDLE  ALT=\"25600=400b%5E6%2Fb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25600=400b%5E4 ALIGN=MIDDLE  ALT=\"25600=400b%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide by subtracting the exponents\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64=b%5E4 ALIGN=MIDDLE  ALT=\"64=b%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 400\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2C64%29=b ALIGN=MIDDLE  ALT=\"root%284%2C64%29=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the fourth root of both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%288%29 ALIGN=MIDDLE  ALT=\"b=sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%2Ab%5Et ALIGN=MIDDLE  ALT=\"y=a%2Ab%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the general exponential equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400=a%2A%28sqrt%288%29%29%5E2 ALIGN=MIDDLE  ALT=\"400=a%2A%28sqrt%288%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2 ALIGN=MIDDLE  ALT=\"t=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%288%29 ALIGN=MIDDLE  ALT=\"b=sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=400 ALIGN=MIDDLE  ALT=\"y=400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400=a%2A8 ALIGN=MIDDLE  ALT=\"400=a%2A8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50=a ALIGN=MIDDLE  ALT=\"50=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 8 to isolate \"a\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So our equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > or if you want an approximation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%282.82843%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%282.82843%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a) \r<BR>\n" );
document.write( "\n" );
document.write( "\"What is the rate of growth of the population of bacteria?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From the equation, the value of \"b\" is the rate of growth. So the rate of growth is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 2.82843\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) \"What was the initial population at time t = 0 hours?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the equation we just found\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%28sqrt%288%29%29%5E0 ALIGN=MIDDLE  ALT=\"y=50%2A%28sqrt%288%29%29%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%281%29 ALIGN=MIDDLE  ALT=\"y=50%2A%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the zeroth power to get 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50 ALIGN=MIDDLE  ALT=\"y=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the initial population is 50\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c) \"Write the function that models the population n(t) after t hours.\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Earlier we found the equation to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%282.82843%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%282.82843%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "d) \"When will the number of bacteria exceed 100,000?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=50%2A%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"y=50%2A%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the equation we just found\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100000=50%2A%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"100000=50%2A%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=100000 ALIGN=MIDDLE  ALT=\"y=100000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000=%28sqrt%288%29%29%5Et ALIGN=MIDDLE  ALT=\"2000=%28sqrt%288%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 50\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000=%288%5E%281%2F2%29%29%5Et ALIGN=MIDDLE  ALT=\"2000=%288%5E%281%2F2%29%29%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"8%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2000=8%5E%28t%2F2%29 ALIGN=MIDDLE  ALT=\"2000=8%5E%28t%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the exponents\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%282000%29%29=log%2810%2C%288%5E%28t%2F2%29%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%282000%29%29=log%2810%2C%288%5E%28t%2F2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the log of both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%282000%29%29=%28t%2F2%29log%2810%2C%288%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%282000%29%29=%28t%2F2%29log%2810%2C%288%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the right side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%282000%29%29%2Flog%2810%2C%288%29%29=%28t%2F2%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%282000%29%29%2Flog%2810%2C%288%29%29=%28t%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%288%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%288%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.65526=t%2F2 ALIGN=MIDDLE  ALT=\"3.65526=t%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use a calculator to evaluate the left side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.31052=t ALIGN=MIDDLE  ALT=\"7.31052=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=7.31052 ALIGN=MIDDLE  ALT=\"t=7.31052\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So it takes about 7.3 hours for the population to exceed 100,000 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );