document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=268781>Question 268781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 15,000, in how many years is the population expected to be 37,500? (Round to the nearest year.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #196966 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=196966\">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=P%28t%29+=+1+%2B+ke%5E%280.12t%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+1+%2B+ke%5E%280.12t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Before we try to answer the question we need to figure out what k is. We are given that the current population is 15,000. The \"t\" for \"current\" can be 0 so our equation for the current population would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000+=+1+%2B+ke%5E%280.12%280%29%29 ALIGN=MIDDLE  ALT=\"15000+=+1+%2B+ke%5E%280.12%280%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This equation has only one unknown, k. So we can use this equation and some Algebra to solve for k. First we simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000+=+1+%2B+ke%5E%280%29 ALIGN=MIDDLE  ALT=\"15000+=+1+%2B+ke%5E%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0+=+1 ALIGN=MIDDLE  ALT=\"e%5E0+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000+=+1+%2B+k ALIGN=MIDDLE  ALT=\"15000+=+1+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 1 from each side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14999+=+k ALIGN=MIDDLE  ALT=\"14999+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now our function is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+1+%2B+14999e%5E%280.12t%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+1+%2B+14999e%5E%280.12t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we are finally ready to answer the question: \"When will the population be 37,500?\"<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37500+=+1+%2B+14999e%5E%280.12t%29 ALIGN=MIDDLE  ALT=\"37500+=+1+%2B+14999e%5E%280.12t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtract 1 from each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37499+=+14999e%5E%280.12t%29 ALIGN=MIDDLE  ALT=\"37499+=+14999e%5E%280.12t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide both sides by 14999<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37499%2F14999+=+e%5E%280.12t%29 ALIGN=MIDDLE  ALT=\"37499%2F14999+=+e%5E%280.12t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we find the natural logarithm (ln) of each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2837499%2F14999%29+=+ln%28e%5E%280.12t%29%29 ALIGN=MIDDLE  ALT=\"ln%2837499%2F14999%29+=+ln%28e%5E%280.12t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2837499%2F14999%29+=+0.12t ALIGN=MIDDLE  ALT=\"ln%2837499%2F14999%29+=+0.12t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we divide both sides by 0.12:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2837499%2F14999%29%2F0.12+=+t ALIGN=MIDDLE  ALT=\"ln%2837499%2F14999%29%2F0.12+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is an exact answer but it is hard to tell what number this is. So we'll get out our calculators and find a decimal approximation of the left side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282.5001000066671111%29%2F0.12+=+t ALIGN=MIDDLE  ALT=\"ln%282.5001000066671111%29%2F0.12+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.9163307337409142%2F0.12+=+t ALIGN=MIDDLE  ALT=\"0.9163307337409142%2F0.12+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.6360894478409515+=+t ALIGN=MIDDLE  ALT=\"7.6360894478409515+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we're told to round to the nearest year, in approximately 8 years the population will be 37,500.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );