document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185949>Question 1185949</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bacteria culture grows with constant relative growth rate .the bacteria count was 200 after 2 hours and 14000 was 5 hours .what is the relative growth rate .what is the initial size of culture . Find the number of cells after 4.5 hours. Find the rate growth after 5.5 hours and when will the population reach 25 000 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849504 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=849504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this exponential growth problem:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Set up the exponential growth formula:**\r<BR>\n" );
document.write( "\n" );
document.write( "The formula for population growth is:  P(t) = P₀ * e^(rt)\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "*   P(t) is the population at time t<BR>\n" );
document.write( "*   P₀ is the initial population<BR>\n" );
document.write( "*   r is the relative growth rate<BR>\n" );
document.write( "*   t is the time\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Use the given information to create two equations:**\r<BR>\n" );
document.write( "\n" );
document.write( "*   Equation 1: 200 = P₀ * e^(2r)  (after 2 hours)<BR>\n" );
document.write( "*   Equation 2: 14000 = P₀ * e^(5r) (after 5 hours)\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Solve for r (relative growth rate):**\r<BR>\n" );
document.write( "\n" );
document.write( "Divide Equation 2 by Equation 1:\r<BR>\n" );
document.write( "\n" );
document.write( "14000 / 200 = (P₀ * e^(5r)) / (P₀ * e^(2r))<BR>\n" );
document.write( "70 = e^(3r)\r<BR>\n" );
document.write( "\n" );
document.write( "Take the natural logarithm of both sides:\r<BR>\n" );
document.write( "\n" );
document.write( "ln(70) = 3r<BR>\n" );
document.write( "r = ln(70) / 3<BR>\n" );
document.write( "r ≈ 1.424 / 3<BR>\n" );
document.write( "r ≈ 0.475 (or 47.5%)\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Solve for P₀ (initial size of the culture):**\r<BR>\n" );
document.write( "\n" );
document.write( "Substitute the value of r back into either Equation 1 or Equation 2. Let's use Equation 1:\r<BR>\n" );
document.write( "\n" );
document.write( "200 = P₀ * e^(2 * 0.475)<BR>\n" );
document.write( "200 = P₀ * e^0.95<BR>\n" );
document.write( "P₀ = 200 / e^0.95<BR>\n" );
document.write( "P₀ ≈ 200 / 2.586<BR>\n" );
document.write( "P₀ ≈ 77\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Find the number of cells after 4.5 hours:**\r<BR>\n" );
document.write( "\n" );
document.write( "P(4.5) = 77 * e^(0.475 * 4.5)<BR>\n" );
document.write( "P(4.5) ≈ 77 * e^2.1375<BR>\n" );
document.write( "P(4.5) ≈ 77 * 8.468<BR>\n" );
document.write( "P(4.5) ≈ 652\r<BR>\n" );
document.write( "\n" );
document.write( "**6. Find the rate of growth after 5.5 hours:**\r<BR>\n" );
document.write( "\n" );
document.write( "The rate of growth is given by the derivative of the population function:\r<BR>\n" );
document.write( "\n" );
document.write( "dP/dt = r * P(t)\r<BR>\n" );
document.write( "\n" );
document.write( "First, find the population at t = 5.5:\r<BR>\n" );
document.write( "\n" );
document.write( "P(5.5) = 77 * e^(0.475 * 5.5) ≈ 77 * 11.85 ≈ 912.45\r<BR>\n" );
document.write( "\n" );
document.write( "Now, find the rate of growth:\r<BR>\n" );
document.write( "\n" );
document.write( "dP/dt = 0.475 * 912.45 ≈ 434 bacteria/hour\r<BR>\n" );
document.write( "\n" );
document.write( "**7. When will the population reach 25,000?**\r<BR>\n" );
document.write( "\n" );
document.write( "25000 = 77 * e^(0.475t)<BR>\n" );
document.write( "25000/77 = e^(0.475t)<BR>\n" );
document.write( "ln(25000/77) = 0.475t<BR>\n" );
document.write( "t = ln(25000/77) / 0.475<BR>\n" );
document.write( "t ≈ ln(324.675) / 0.475<BR>\n" );
document.write( "t ≈ 5.78 / 0.475<BR>\n" );
document.write( "t ≈ 12.17 hours\r<BR>\n" );
document.write( "\n" );
document.write( "**Summary of Answers:**\r<BR>\n" );
document.write( "\n" );
document.write( "*   Relative growth rate (r) ≈ 47.5% or 0.475<BR>\n" );
document.write( "*   Initial size of the culture (P₀) ≈ 77<BR>\n" );
document.write( "*   Number of cells after 4.5 hours ≈ 652<BR>\n" );
document.write( "*   Rate of growth after 5.5 hours ≈ 434 bacteria/hour<BR>\n" );
document.write( "*   Time to reach 25,000 cells ≈ 12.17 hours<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );