document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039910>Question 1039910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The logistic growth function f(t) =50,000/1 + 1249.0e-1.3t<BR>\n" );
document.write( "models the number of people who have become ill with a particular infection t weeks after its initial outbreak in a particular community. What is<BR>\n" );
document.write( "the limiting size of the population that becomes ill? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654738 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=654738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you don't insert parentheses where they belong then you run into trouble.\r<BR>\n" );
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document.write( "you equation needs to be:<BR>\n" );
document.write( "f(t) = 50000/(1+1249*e^(-1.3t))\r<BR>\n" );
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document.write( "note the parentheses around the exponent.<BR>\n" );
document.write( "this insures the entire expression of -1.3t is treated as part of the exponent of e.\r<BR>\n" );
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document.write( "note the parentheses around (1+1249*e^(-1.3t))<BR>\n" );
document.write( "this insures the entire expression of (1+1249*e^(-1.3t)) is included in the denominator.\r<BR>\n" );
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document.write( "you can graph this equation by replacing f(t) with y and replacing t with x.\r<BR>\n" );
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document.write( "then you can see what the equation looks like.\r<BR>\n" );
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document.write( "you have to be a little creative with the dimensions of the graph.<BR>\n" );
document.write( "after some fiddling, i settled on the range of the x-axis being -15 to 25, and the range of the y-axis being -1000 to 100,000.\r<BR>\n" );
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document.write( "this allowed me to see what was going on.\r<BR>\n" );
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document.write( "what was going on is that the equation starts off small and then rises to about 50000 and then flattens out and doesn't go any higher than 50,000.\r<BR>\n" );
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document.write( "replacing x with some selected values and calculating y for those values yields a table similar to what you see below:\r<BR>\n" );
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document.write( "x        y\r\n" );
document.write( "-3       .81031\r\n" );
document.write( "-2       2.9731\r\n" );
document.write( "-1       10.908\r\n" );
document.write( "0        40\r\n" );
document.write( "1        146\r\n" );
document.write( "2        533\r\n" );
document.write( "3        1902\r\n" );
document.write( "4        6337\r\n" );
document.write( "5        177374\r\n" );
document.write( "6        33074\r\n" );
document.write( "7        43880\r\n" );
document.write( "8        48169\r\n" );
document.write( "9        49487\r\n" );
document.write( "10       49859\r\n" );
document.write( "...\r\n" );
document.write( "15       50000\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "by the time x gets to 15, it has saturated to y = 50,000 which is the limiting value of the function.\r<BR>\n" );
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document.write( "the above values have been rounded to the nearest integer.\r<BR>\n" );
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document.write( "this is caused by the denominator of (1 + 1249 * e^(-1.3*x) becoming very close to 1 as x gets larger than 15; close enough that the numerator of 50,000 is effectively being divided by 1.\r<BR>\n" );
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document.write( "for example, (1 + 1249*e^(-1.3*x) is equal to (1 + .0000042444...) which becomes 1.0000042444 which is pretty darn close to 1.\r<BR>\n" );
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document.write( "50000 divided by 1.0000042444 is equal to 49999.787778<BR>\n" );
document.write( "that is very close to 50,000 and gets even closer as x gets larger than 15.\r<BR>\n" );
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document.write( "when x = 20, the function becomes y = 49999.99968.\r<BR>\n" );
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document.write( "that even closer.\r<BR>\n" );
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document.write( "when you round the result to the nearest integer, you get 50,000 for both x = 15 and x = 20.\r<BR>\n" );
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document.write( "bottom line:\r<BR>\n" );
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document.write( "don't forget to use parentheses to ensure the function is being displayed correctly.\r<BR>\n" );
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document.write( "take some spot measurements if you don't have a graphing calculator so you can see what is happening.\r<BR>\n" );
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document.write( "the limiting value of your function is 50,000.\r<BR>\n" );
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document.write( "my graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/070301.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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