document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1065322>Question 1065322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of <BR>\n" );
document.write( "8.7% per hour. How many hours does it take for the size of the sample to double? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #680487 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=680487\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> continuoous growth model 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 2.718281828.....<BR>\n" );
document.write( "r is the rate of growth 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:\r<BR>\n" );
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document.write( "f = 2<BR>\n" );
document.write( "p = 1<BR>\n" );
document.write( "r = .087 per hour.<BR>\n" );
document.write( "t = number of hours.\r<BR>\n" );
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document.write( "formula becomes 2 = 1 * e^(.087 * t)\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(2) = ln(1 * e^(.087 * t)\r<BR>\n" );
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document.write( "this becomes ln(2) = ln(1) + .087 * t * ln(e)\r<BR>\n" );
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document.write( "since ln(1) = 0 and ln(e) = 1, this becomes ln(2) = .087 * t\r<BR>\n" );
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document.write( "solve for t to get t = ln(2) / .087 = 7.967208972\r<BR>\n" );
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document.write( "the sample should double in 7.967208972 hours.\r<BR>\n" );
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document.write( "replace t with 7.967208972 in your original equaiton and you get:\r<BR>\n" );
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document.write( "2 = 1 * e^(.087 * 7.967208972).\r<BR>\n" );
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document.write( "evaluate this equation to get 2 = 2\r<BR>\n" );
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document.write( "this confirms the solution is correct.\r<BR>\n" );
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document.write( "note that:\r<BR>\n" );
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document.write( "ln(1) = 0<BR>\n" );
document.write( "ln(e) = 1<BR>\n" );
document.write( "ln(a * b^c) = ln(a) + ln(b^c) which is then equal to ln(a) + c*ln(b)\r<BR>\n" );
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document.write( "you can use your calculator to confirm the first 2 statements are true.<BR>\n" );
document.write( "the third statement is based on the properties of logarithms.\r<BR>\n" );
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document.write( "here's a reference that might help.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/logrules.htm\" target=\"_blank\">http://www.purplemath.com/modules/logrules.htm</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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