document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1154017>Question 1154017</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The types of habitat required for nesting limits the population of a certain species of bird. The population (in billions) behaves according to the “logistic growth” model given by:  <BR>\n" );
document.write( "P(t) = Ae^(0.01t)/2+e^(0.01t)\r<BR>\n" );
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document.write( "where t is the number of years from today and A is a constant.\r<BR>\n" );
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document.write( "(a)If the population is 7billion today, how long will it take to<BR>\n" );
document.write( "reach 8 billion? Round your final answer to the nearest integer.\r<BR>\n" );
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document.write( "b)What eventually happens to the population of this bird species?<BR>\n" );
document.write( "Hint: To help simplify the expression consider dividing each<BR>\n" );
document.write( "term by something </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #776392 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13209)</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=776392\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "You show the function as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+Ae%5E%280.01t%29%2F2%2Be%5E%280.01t%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+Ae%5E%280.01t%29%2F2%2Be%5E%280.01t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Clearly that is not a logistic function.  Use parentheses where they are required!!<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+Ae%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+Ae%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The current population (t=0) in billions is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29+=+A%2A1%2F%282%2B1%29+=+A%2F3+=+7 ALIGN=MIDDLE  ALT=\"P%280%29+=+A%2A1%2F%282%2B1%29+=+A%2F3+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+21 ALIGN=MIDDLE  ALT=\"A+=+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+21e%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29 ALIGN=MIDDLE  ALT=\"P%28t%29+=+21e%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(a) Find the number of years until the population reaches 8 billion.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+=+Ae%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29 ALIGN=MIDDLE  ALT=\"8+=+Ae%5E%280.01t%29%2F%282%2Be%5E%280.01t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That can be solved algebraically, but it is very tedious.  Use a graphing calculator to graph the function itself and the constant 8 and find the intersection.  It is at t=20.764 to 3 decimal places.<br><BR>\n" );
document.write( "ANSWER: 21 years, according to the instructions to give the answer as the nearest integer.<br><BR>\n" );
document.write( "(b) What eventually happens to the population?<br><BR>\n" );
document.write( "As t gets large, the \"2\" in the denominator becomes insignificant; the function approaches<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29+=+21e%5E%280.01t%29%2F%28e%5E%280.01t%29%29+=+21 ALIGN=MIDDLE  ALT=\"P%28t%29+=+21e%5E%280.01t%29%2F%28e%5E%280.01t%29%29+=+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: In a long period of time, the limit of the population is 21 billion.<br><BR>\n" );
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