document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=159133>Question 159133</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the function P = 13 + 14 ln x, represents the perecentage of inbound e-mail in the US. that is considered Spam, where x represents the number of years after 2000.<BR>\n" );
document.write( "Use this model to determine in how many years(to two decimal places) it will take for the percentage to reach 95%, provided that law enforcement regarding spammers does not change. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #117371 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=117371\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> P = percentage = 95<BR>\n" );
document.write( "13+14*ln(x) = 95<BR>\n" );
document.write( "ln(x) = (95-13)/14<BR>\n" );
document.write( "ln(x) = 5.857142857<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ln%28x%29 ALIGN=MIDDLE  ALT=\"y+=+ln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ey+=+x ALIGN=MIDDLE  ALT=\"e%5Ey+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y = ln(x) = 5.857142857, so y = 5.857142857<BR>\n" );
document.write( "equation for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ey+=+x ALIGN=MIDDLE  ALT=\"e%5Ey+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%285.857142857%29+=+x ALIGN=MIDDLE  ALT=\"e%5E%285.857142857%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "use the calculator to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%285.857142857%29 ALIGN=MIDDLE  ALT=\"e%5E%285.857142857%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "answer becomes x = 349.7235051<BR>\n" );
document.write( "substituting in the original equation of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95+=+13+%2B+%2814+%2A+ln%28x%29%29 ALIGN=MIDDLE  ALT=\"95+=+13+%2B+%2814+%2A+ln%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95+=+13+%2B+%2814+%2A+ln%28349.7235051%29%29 ALIGN=MIDDLE  ALT=\"95+=+13+%2B+%2814+%2A+ln%28349.7235051%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "solving for ln(349.7235051) using the calculator, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95+=+13+%2B+%2814+%2A+5.857142857%29 ALIGN=MIDDLE  ALT=\"95+=+13+%2B+%2814+%2A+5.857142857%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=95+=+13+%2B+82 ALIGN=MIDDLE  ALT=\"95+=+13+%2B+82\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which becomes<BR>\n" );
document.write( "95 = 95.<BR>\n" );
document.write( "answer is 349.72 years rounding to the nearest hundredth.<BR>\n" );
document.write( "now that we know how far out x has to go to satisfy the equation, we can graph it as follows:<BR>\n" );
document.write( "range of x = -1 to 400 where x represents years from 2000.<BR>\n" );
document.write( "range of y = -10 to 100 where y represents percent.<BR>\n" );
document.write( "-----------------------------------------------------<BR>\n" );
document.write( "graph looks like this<BR>\n" );
document.write( "please scan below the graph for further comments.<BR>\n" );
document.write( "-----------------------------------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281200%2C1200%2C-100%2C400%2C-50%2C100%2C%2814%2Aln%28x%29%29%2B13%29 ALIGN=MIDDLE  ALT=\"graph%281200%2C1200%2C-100%2C400%2C-50%2C100%2C%2814%2Aln%28x%29%29%2B13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------------------------------------------<BR>\n" );
document.write( "range of x had to start from - something in order for the x-axis to display properly.<BR>\n" );
document.write( "range of y had to start from - something in order for the y-axis to display properly<BR>\n" );
document.write( "the formula of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+14%2Aln%28x%29%2B13 ALIGN=MIDDLE  ALT=\"y+=+14%2Aln%28x%29%2B13\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not defined for x <= 0 because e^y will never be negative and e^y can approach 0 but never be 0.<BR>\n" );
document.write( "--------------------------------------------<BR>\n" );
document.write( "based on the formula as shown, x = 349.72 rounded to the nearest hundredth of a year.<BR>\n" );
document.write( "that year will be 2000 + 349.72 = 2349.72 based on the formula provided.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );