document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099143>Question 1099143</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of people in a community who hear a particular rumor is governed by the<BR>\n" );
document.write( "logistic model<BR>\n" );
document.write( "R(t) =<BR>\n" );
document.write( "L 1 + Me&#8722;&#954;t where R(t) represents the number of people who have heard the rumor t days after the<BR>\n" );
document.write( "rumor initially was told by one person to another person. The constants L, M and &#954;<BR>\n" );
document.write( "are to be determined from observations of the spreading rumor.<BR>\n" );
document.write( "The following observations of the rumor “The cake is a lie” spreading across a uni<BR>\n" );
document.write( "versity campus are made • Initially one student tells another student the rumor, that is, only 2 students have heard the rumor. • The rumor spreads to 10 students after 1 day. • After a number of days of observations it is predicted that eventually, that is, after a very long time, 200 students on campus will have heard the rumor.<BR>\n" );
document.write( "Given the three pieces of information provided from the observations of the rumor, &#64257;nd<BR>\n" );
document.write( "the exact values of the constants L, M and &#954;. Write the equation describing how many<BR>\n" );
document.write( "people have heard the rumor at time t days after the rumor starts </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713646 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=713646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The logistic function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+L%2F%281%2BMe%5E%28-kt%29%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+L%2F%281%2BMe%5E%28-kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The numerator \"L\" is the limiting value of the function, as t goes to infinity.  This is because as t gets very large, the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-kt%29 ALIGN=MIDDLE  ALT=\"e%5E%28-kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes to 0, making the function value L/1 = L.<br><BR>\n" );
document.write( "So in your problem, L is 200, and the function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+200%2F%281%2BMe%5E%28-kt%29%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+200%2F%281%2BMe%5E%28-kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The function value at t=0 is the starting value of the function.  The function evaluated at t=0 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%2F%281%2BM%29 ALIGN=MIDDLE  ALT=\"200%2F%281%2BM%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because when t=0 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-kt%29+=+1 ALIGN=MIDDLE  ALT=\"e%5E%28-kt%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "In your problem, where the initial number of people who have heard the rumor is 1, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%2F%281%2BM%29+=+1 ALIGN=MIDDLE  ALT=\"200%2F%281%2BM%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means M is 199.<br><BR>\n" );
document.write( "So the function in this problem is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+200%2F%281%2B199e%5E%28-kt%29%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+200%2F%281%2B199e%5E%28-kt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find the value of k to finish the problem, we use the given information that 10 students have heard the rumor after 1 day.  That is, R(1) = 10.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%281%29+=+10+=+200%2F%281%2B199e%5E%28-k%29%29 ALIGN=MIDDLE  ALT=\"R%281%29+=+10+=+200%2F%281%2B199e%5E%28-k%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B199e%5E%28-k%29+=+200%2F10+=+20 ALIGN=MIDDLE  ALT=\"1%2B199e%5E%28-k%29+=+200%2F10+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=199e%5E%28-k%29+=+19 ALIGN=MIDDLE  ALT=\"199e%5E%28-k%29+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-k%29+=+19%2F199 ALIGN=MIDDLE  ALT=\"e%5E%28-k%29+=+19%2F199\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now in the denominator of the logistic function, notice that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-kt%29+=+%28e%5E%28-k%29%29%5Et%29 ALIGN=MIDDLE  ALT=\"e%5E%28-kt%29+=+%28e%5E%28-k%29%29%5Et%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since we have found that the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-k%29 ALIGN=MIDDLE  ALT=\"e%5E%28-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%2F199 ALIGN=MIDDLE  ALT=\"19%2F199\"   DISABLED_style= \"max-width:90%; height:auto;\" >; we can now write the completed function for this problem as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+200%2F%281%2B199%2819%2F199%29%5Et%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+200%2F%281%2B199%2819%2F199%29%5Et%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With the logistic function in this form, it is easy to see that the given three data points are satisfied:<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%280%29+=+200%2F%281%2B199%281%29%29+=+200%2F200+=+1 ALIGN=MIDDLE  ALT=\"R%280%29+=+200%2F%281%2B199%281%29%29+=+200%2F200+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%281%29+=+200%2F%281%2B199%2819%2F199%29%29+=+200%2F%281%2B19%29+=+200%2F20+=+10 ALIGN=MIDDLE  ALT=\"R%281%29+=+200%2F%281%2B199%2819%2F199%29%29+=+200%2F%281%2B19%29+=+200%2F20+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) For t \"large\", <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+200%2F%281%2B199%2A0%29+=+200%2F1+=+200 ALIGN=MIDDLE  ALT=\"R%28t%29+=+200%2F%281%2B199%2A0%29+=+200%2F1+=+200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now I note that the way you have asked the question, we are supposed to find the values of L, M, and k.  Finding the values of L and M was easy.<br><BR>\n" );
document.write( "But we didn't find the value of k in the work above.  However, that is easy:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-k%29+=+19%2F199 ALIGN=MIDDLE  ALT=\"e%5E%28-k%29+=+19%2F199\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-k+=+ln%2819%2F199%29 ALIGN=MIDDLE  ALT=\"-k+=+ln%2819%2F199%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-ln%2819%2F199%29 ALIGN=MIDDLE  ALT=\"k+=+-ln%2819%2F199%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We could write the logistic function using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-kt%29 ALIGN=MIDDLE  ALT=\"e%5E%28-kt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >; but it would look ugly:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+200%2F%281%2B199%28e%5E%28%28-ln19%2F199%29%2At%29%29%29 ALIGN=MIDDLE  ALT=\"R%28t%29+=+200%2F%281%2B199%28e%5E%28%28-ln19%2F199%29%2At%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );