document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=678697>Question 678697</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Exponential growth problem using the formula: N= No(R)^t/d <BR>\n" );
document.write( "N is the current amount, No is the initial amount, R is the rate, t is the time, d is the number of days/months/years \r<BR>\n" );
document.write( "\n" );
document.write( "The population of a city was estimated to be 125000 in 1930 and 500000 in 1998.<BR>\n" );
document.write( "a) Estimate the population of the city in 2020. <BR>\n" );
document.write( "b) if the population continues to grow at the same rate, when will the population reach 1 million?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #421636 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=421636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > grows exponentially,<BR>\n" );
document.write( "we know the \"initial\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5B0%5D ALIGN=MIDDLE  ALT=\"N%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > at some point we take as starting point,<BR>\n" );
document.write( "and we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > grew by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > over a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > days/months/years (or whatever unit of time),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28N=N%5B0%5D%2AR%5E%28%28t%2Fd%29%29%29%5Efunction ALIGN=MIDDLE  ALT=\"highlight%28N=N%5B0%5D%2AR%5E%28%28t%2Fd%29%29%29%5Efunction\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that gives the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > at a time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > days/months/years (or whatever unit of time we were using) after our starting point.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE PROBLEM SET-UP:<BR>\n" );
document.write( "Here we define<BR>\n" );
document.write( "start = year 1930<BR>\n" );
document.write( "we measure time in years<BR>\n" );
document.write( "<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;\" > for 1930<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1998-1930=68 ALIGN=MIDDLE  ALT=\"t=1998-1930=68\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 1998<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=2020-1930=90 ALIGN=MIDDLE  ALT=\"t=2020-1930=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 2020<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=68 ALIGN=MIDDLE  ALT=\"d=68\"   DISABLED_style= \"max-width:90%; height:auto;\" > years (between 1930 and 1998)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=500000%2F125000=4 ALIGN=MIDDLE  ALT=\"R=500000%2F125000=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the growth factor over 68 years<BR>\n" );
document.write( "(the population quadrupled in the 68 years between 1930 and 1998).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5B0%5D=125000 ALIGN=MIDDLE  ALT=\"N%5B0%5D=125000\"   DISABLED_style= \"max-width:90%; height:auto;\" > is our starting population<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28N=125000%2A4%5E%28%28t%2F68%29%29%29%5Efunction ALIGN=MIDDLE  ALT=\"highlight%28N=125000%2A4%5E%28%28t%2F68%29%29%29%5Efunction\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=years ALIGN=MIDDLE  ALT=\"t=years\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=after ALIGN=MIDDLE  ALT=\"after\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1930 ALIGN=MIDDLE  ALT=\"1930\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "PART a:<BR>\n" );
document.write( "In the year 2020, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=90 ALIGN=MIDDLE  ALT=\"t=90\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the population is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28N=125000%2A4%5E%28%2890%2F68%29%29%29%5E%28a%2Acalculation%29 ALIGN=MIDDLE  ALT=\"highlight%28N=125000%2A4%5E%28%2890%2F68%29%29%29%5E%28a%2Acalculation%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28N=1782986%29%5E%28a%28approx%29%29 ALIGN=MIDDLE  ALT=\"highlight%28N=1782986%29%5E%28a%28approx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "PART b:<BR>\n" );
document.write( "To calculate when the population will be one million.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281000000=125000%2A4%5E%28%28t%2F68%29%29%29%5E%28b%2Acalculation%29 ALIGN=MIDDLE  ALT=\"highlight%281000000=125000%2A4%5E%28%28t%2F68%29%29%29%5E%28b%2Acalculation%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Taking logs of both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281000000%29=log%28125000%29%2B%28t%2F68%29%2Alog%284%29 ALIGN=MIDDLE  ALT=\"log%281000000%29=log%28125000%29%2B%28t%2F68%29%2Alog%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From there we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28t%2F68=3%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28t%2F68=3%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28t=102%29 ALIGN=MIDDLE  ALT=\"highlight%28t=102%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which corresponds to year <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1930%2B102=highlight%282032%29 ALIGN=MIDDLE  ALT=\"1930%2B102=highlight%282032%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "I really did not bother with the calculation.<BR>\n" );
document.write( "It was just mental math.<BR>\n" );
document.write( "I knew (from way before) that the population had quadrupled in 68 years.<BR>\n" );
document.write( "That meant that it doubled every 34 years,<BR>\n" );
document.write( "so that in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=68=34%2A2 ALIGN=MIDDLE  ALT=\"68=34%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > years it would double twice,<BR>\n" );
document.write( "and would end up multiplying times 2 twice to quadruple.<BR>\n" );
document.write( "Then, being 500,000 in 1998,<BR>\n" );
document.write( "it needed just another 32 years to double again, and reach one million. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );