document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186153>Question 1186153</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a limited environment where A is the maximum number of bacteria supportable by the environment, the rate of bacterial growth is directly proportional to the number present and the difference between A and the number present. Suppose 1 million bacteria is the maximum number supportable by the environment and the rate of growth is 60 bacteria per minute when there are 1000 bacteria present.\r<BR>\n" );
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document.write( "Find the rate of growth when there are 100,000 bacteria present. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849475 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849475\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**1. Set up the differential equation:**\r<BR>\n" );
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document.write( "Let *B(t)* be the number of bacteria at time *t*. The problem states that the rate of growth is directly proportional to the number present and the difference between A and the number present.  This can be written as:\r<BR>\n" );
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document.write( "dB/dt = k * B * (A - B)\r<BR>\n" );
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document.write( "where *k* is the proportionality constant.\r<BR>\n" );
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document.write( "**2. Use the given information to find *k*:**\r<BR>\n" );
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document.write( "We know that A = 1,000,000, and when B = 1000, dB/dt = 60.  Plugging these values into the equation:\r<BR>\n" );
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document.write( "60 = k * 1000 * (1,000,000 - 1000)<BR>\n" );
document.write( "60 = k * 1000 * 999,000<BR>\n" );
document.write( "k = 60 / (1000 * 999,000)<BR>\n" );
document.write( "k = 6 / 999,000<BR>\n" );
document.write( "k ≈ 6.006 x 10⁻⁸\r<BR>\n" );
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document.write( "**3. Find the rate of growth when B = 100,000:**\r<BR>\n" );
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document.write( "Now we want to find dB/dt when B = 100,000.  Using the same equation and the calculated value of *k*:\r<BR>\n" );
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document.write( "dB/dt = (6.006 x 10⁻⁸) * 100,000 * (1,000,000 - 100,000)<BR>\n" );
document.write( "dB/dt = (6.006 x 10⁻⁸) * 100,000 * 900,000<BR>\n" );
document.write( "dB/dt = 5405.4 bacteria per minute\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The rate of growth when there are 100,000 bacteria present is approximately 5405.4 bacteria per minute.<BR>\n" );
document.write( " </SPAN>\n" );
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