document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122273>Question 1122273</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At noon a culture of bacteria had 2.5x10^6 members, and at 3 pm the population was 4.5x10^6. Assuming exponential growth, find when the population will be 8.0x10^6. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #738356 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=738356\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you first need to find the exponential growth rate.\r<BR>\n" );
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document.write( "the formula for that is f = p * e^(rt)\r<BR>\n" );
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document.write( "f is equal to 4.5 * 10^6<BR>\n" );
document.write( "p is equal to 1.5 * 10^6<BR>\n" );
document.write( "t is equal to 3\r<BR>\n" );
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document.write( "you are solving for r.\r<BR>\n" );
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document.write( "formula becomes 4.5 * 10^6 = 1.5 * 10^6 * e^(3r)\r<BR>\n" );
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document.write( "divide both sides of this formula by 1.5 * 10^6 to get:\r<BR>\n" );
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document.write( "4.5 * 10^6 / (1.5 * 10^6) = e^(3r)\r<BR>\n" );
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document.write( "take the natural log of both sides of this formula and simplify to get:\r<BR>\n" );
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document.write( "ln(3) = ln(e^3r)\r<BR>\n" );
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document.write( "since ln(e^3r) is equal to 3r * ln(e) and ln(e) is equal to 1, this equation becomes:\r<BR>\n" );
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document.write( "ln(3) = 3r\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = ln(3) / 3\r<BR>\n" );
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document.write( "this results in r = .3662040962\r<BR>\n" );
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document.write( "that's your hourly exponential growth rate.\r<BR>\n" );
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document.write( "to see if this is good, take 1.5 * 10^6 and multiply it by e^(.3662040962 * 3).\r<BR>\n" );
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document.write( "you will get 4.5 * 10^6 which is exactly what you want, assuming the rate is calculated correctly, as it is.\r<BR>\n" );
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document.write( "to find out when the population will reach 8.0 * 10^6, use the same formula of f = p * e^(rt).\r<BR>\n" );
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document.write( "if you are starting from 1.5 * 10^6, the formula becomes:\r<BR>\n" );
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document.write( "8.0 * 10^6 = 1.5 * 10^6 * e^(.3662040962 * t)\r<BR>\n" );
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document.write( "divide both sides of the equation by 1.5 * 10^6 to get:\r<BR>\n" );
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document.write( "8.0 * 10^6 / (1.5 * 10^6) = e^(.3662040962 * t)\r<BR>\n" );
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document.write( "take the natural log of both sides and simplify to get:\r<BR>\n" );
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document.write( "ln(5 + 1/3) = .3662040962 * t\r<BR>\n" );
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document.write( "solve for t to get:\r<BR>\n" );
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document.write( "t = ln(5 + 1/3) / .3662040962 = 4.571157043\r<BR>\n" );
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document.write( "to confirm, take 1.5 * 10^6 and multiply it by e^(.3662040962 * 4.571157043).\r<BR>\n" );
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document.write( "you will get 8.0 * 10^6, as you should.\r<BR>\n" );
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document.write( "if you had started from 4.5 * 10^6, the formula would have become:\r<BR>\n" );
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document.write( "8.0 * 10^6 = 4.5 * 10^6 * e^(.3662040962 * t)\r<BR>\n" );
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document.write( "in that case, you would have solved for t to get:\r<BR>\n" );
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document.write( "t = ln(8.0 * 10^6 / (4.5 * 10^6) / .3662040962, resulting in:\r<BR>\n" );
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document.write( "t = 1.571157043\r<BR>\n" );
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document.write( "add that to the 3 hours to get from 1.5 * 10^6 to 4.5 * 10^6 and the total hours is 4.571157043.\r<BR>\n" );
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document.write( "the total hours to get from 1.5 to 4.5 million = 3.\r<BR>\n" );
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document.write( "the total hours to get from 4.5 to 8 million = 1.571157043\r<BR>\n" );
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document.write( "the total hours to get from 1.5 to 8 million = 4.571157043\r<BR>\n" );
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document.write( "the formula for this problem can be graphed as shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/090201.jpg\" alt=\"$$$\">\r<BR>\n" );
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document.write( " </SPAN>\n" );
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