document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=920739>Question 920739</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an exponential growth model for the following scenario.  A bacterial strain was measured to have a total population of 400 after 2 hours.  After 5 hours the population was 900.  Use this model to predict the population after 10 hours. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #558574 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=558574\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe the exponential growth model equation is f = p * e^(rt)\r<BR>\n" );
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document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "e is the scientific constant of 2.71828...<BR>\n" );
document.write( "r is the growth rate per time period.<BR>\n" );
document.write( "t is the number of time periods.\r<BR>\n" );
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document.write( "in your problem, the bacterial grow from 400 to 900 in 3 hours.\r<BR>\n" );
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document.write( "the population is 400 after 2 hours and the population is 900 after 5 hours.\r<BR>\n" );
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document.write( "that's 3 hours in between.\r<BR>\n" );
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document.write( "your equation becomes:\r<BR>\n" );
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document.write( "900 = 400 * e^(3r)\r<BR>\n" );
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document.write( "f = 900<BR>\n" );
document.write( "p = 400<BR>\n" );
document.write( "t = 3 hours<BR>\n" );
document.write( "r = what you want to solve for.\r<BR>\n" );
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document.write( "divide both sides of this equation by 400 and you get:\r<BR>\n" );
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document.write( "900/400 = e^(3r)\r<BR>\n" );
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document.write( "simplify to get 9/4 = e^(3r)\r<BR>\n" );
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document.write( "take the natural log of both sides of this equation to get:\r<BR>\n" );
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document.write( "ln(9/4) = ln(e^(3r))\r<BR>\n" );
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document.write( "since ln(e^(3r)) equals 3r * ln(e) and ln(e) = 1, you get:\r<BR>\n" );
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document.write( "ln(9/4) = 3r\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "ln(9/4) / 3 = r\r<BR>\n" );
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document.write( "solve for r to get r = ln(9/4) / 3 = .2703100721\r<BR>\n" );
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document.write( "confirm by replacing r with .2703100721 in the original equation to get:\r<BR>\n" );
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document.write( "900 = 400 * e^(.2703100721*3) which becomes:\r<BR>\n" );
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document.write( "900 = 900.\r<BR>\n" );
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document.write( "this confirms the solution is good.\r<BR>\n" );
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document.write( "you are asked to find how many bacteria after 10 hours.\r<BR>\n" );
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document.write( "the formula for that is the same general formula of f = p * e^(rt)\r<BR>\n" );
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document.write( "you can choose p to be 400 or 900<BR>\n" );
document.write( "if you choose 400, then t = 10-2 = 8 hours<BR>\n" );
document.write( "if you choose 900, then t = 10 - 5 = 5 hours<BR>\n" );
document.write( "you should get the same answer.<BR>\n" );
document.write( "i'll do both to show you.\r<BR>\n" );
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document.write( "when p = 400<BR>\n" );
document.write( "t = 10-2 = 8<BR>\n" );
document.write( "r = .2703100721<BR>\n" );
document.write( "f = what you want to find.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "f = 400 * e^(.2703100721 * 8) = 3477.069512\r<BR>\n" );
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document.write( "when p = 900<BR>\n" );
document.write( "t = 10-5 = 5<BR>\n" );
document.write( "r = .2703100721<BR>\n" );
document.write( "f = what you want to find.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "f = 900 * e^(.2703100721 * 5 = 3477.069512\r<BR>\n" );
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document.write( "you're good either way.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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