document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027952>Question 1027952</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population growth follows a logistic growth model. <BR>\n" );
document.write( "p(t)=d/1+ke^-ct\r<BR>\n" );
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document.write( "where c,d,and k are positive constants. For a certain fish population in a small pond d= 1200, k= 11, c = 0.2, and t is measured in years. The fish were introduced into the pond at time = 0. <BR>\n" );
document.write( "a)<BR>\n" );
document.write( "How many fish were originally put into the pond? <BR>\n" );
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document.write( "Find the population of fish <BR>\n" );
document.write( "after 10, 20, and 30 years. <BR>\n" );
document.write( "c)<BR>\n" );
document.write( "Evaluate P(t) for large values of t. What value does the population approach as <BR>\n" );
document.write( " t&#8594;&#8734;? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #643139 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=643139\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it appears that the formula is:\r<BR>\n" );
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document.write( "p(t) = d/(1+ke^-ct)\r<BR>\n" );
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document.write( "that set of parentheses is very important because it changes the normal order of operations.\r<BR>\n" );
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document.write( "using the normal order of operations without parentheses, you would get:\r<BR>\n" );
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document.write( "p(t) = (d/1) + (ke^-ct)\r<BR>\n" );
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document.write( "that doesn't make a lot of sense because you would be getting no growth at all.\r<BR>\n" );
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document.write( "i'm assuming the formula is p(t) = d/(1+ke^-ct)\r<BR>\n" );
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document.write( "under that assumption, the population will grow to 1200.\r<BR>\n" );
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document.write( "when d = 1200 and k = 11 and c = .2, you get:\r<BR>\n" );
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document.write( "when t = 0, p(0) = 1200 / (1 + 11 * e^(-.2*0) which becomes p(0) = 1200 / (1 + 11 * e^0) which becomes p(0) = 1200 / (1 + 11) which becomes p(0) = 1200 / 12 which becomes p(0) = 100\r<BR>\n" );
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document.write( "the initial number of fish is therefore 100.\r<BR>\n" );
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document.write( "when t = 10, p(10) = 1200 / (1 + 11 * e^(-.2*10) which becomes p(10) = 1200 / (1 + 11 * e^-2) which becomes p(10) = 1200 / (1 + 1.488688116) which becomes p(10) = 1200 / (2.488688116) which becomes p(10) = 482 rounded to the nearest integer.\r<BR>\n" );
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document.write( "when t = 20, p(20) = 1200 / (1 + 11 * e^(-.2*20) which becomes p(20) = 1200 / (1 + 11 * e^-4) which becomes p(20) = 1200 / (1 + .2014720278) which becomes p(20) = 1200 / (1.2014720278) which becomes p(20) = 999 rounded to the nearest integer.\r<BR>\n" );
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document.write( "when t = 30, p(30) = 1200 / (1 + 11 * e^(-.2*30) which becomes p(30) = 1200 / (1 + 11 * e^(-6) which becomes p(20) = 1200 / (1 + .0024787522) which becomes p(30) = 1200 / 1.0024787522) which becomes p(30) = 1197 rounded to the nearest integer.\r<BR>\n" );
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document.write( "it is starting to become clear that, as t gets larger, e^(-.2*t) becomes smaller.  \r<BR>\n" );
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document.write( "this leads to the conclusion as t approaches infinity, e^(-.2*t) will approach 0.\r<BR>\n" );
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document.write( "p(infinity) will therefore becomes 1200 / (1 + 0) which becomes 1200 / 1 which becomes 1200.\r<BR>\n" );
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document.write( "the number of fish in the pond will saturate at 1200, given that d = 1200 and k = 11 and c = .2 in the formula of p(t) = d / (1 + e^(-ct))\r<BR>\n" );
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