document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=387793>Question 387793</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A biologist finds that the population of a certain type of bacteria doubles each half-hour. An initial culture has 90 bacteria.<BR>\n" );
document.write( "How long will it take for the number of bacteria to reach 368,640? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #275030 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=275030\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> first, let us make a table to see how the data behaves:\r<BR>\n" );
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document.write( "1/2  hour periods                                   0          1          2          3          4 .......\r<BR>\n" );
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document.write( "number of bacteria                              90        180      360    720     1440\r<BR>\n" );
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document.write( "As stated in the problem, the above shows that the initial number of bacteria doubles each 1/2 hour. It can also be seen that we can calculate the number of bacteria reached at any period by multiplying the original amount of 90 by 2^period number. For example, the 720 reached after 3 periods is determined by multiplying the original amount of 90 by 2^3, after 4 periods by 2^4, after 2 periods by 2^2, etc.   \r<BR>\n" );
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document.write( "If we call the original amount, P, and the amount reached after a certain number of 1/2 hour periods, A, and period number, n, we can come up with the following relationship:\r<BR>\n" );
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document.write( "A =  P (2^n)\r<BR>\n" );
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document.write( "For this problem:\r<BR>\n" );
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document.write( "368640 = 90(2^n)<BR>\n" );
document.write( "2^n=368640/90=4096<BR>\n" );
document.write( "use logarithms to solve<BR>\n" );
document.write( "n (log 2)=log 4096<BR>\n" );
document.write( "n=log 4096/(log 2) = 12\r<BR>\n" );
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document.write( "ans: after 12 (1/2 hour) periods, or 6 hours, the bacteria count would have reached 368640 </SPAN>\n" );
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