document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188652>Question 1188652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bacteria culture starts with 280 bacteria and grows at a rate proportional to its size. After 5 hours there will be 1400 bacteria. \r<BR>\n" );
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document.write( "A. What will the population be after 4 hours?\r<BR>\n" );
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document.write( "B. How long will it take the population to reach 2630? Give your answer accurate to 2 decimal places.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819790 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=819790\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think this means that it grows at a constant rate.<BR>\n" );
document.write( "for example, something like 10% of its size, no matter what its size is.\r<BR>\n" );
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document.write( "the formula for that is f = p * (1 + r) ^ n\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( "r is the rate of growth per time period.<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "they state that it grows from 280 to 1400 in 5 hours.\r<BR>\n" );
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document.write( "the formula becomes 1400 = 280 * (1 + r) ^ 5.<BR>\n" );
document.write( "divide both sides of this formula by 280 to get:<BR>\n" );
document.write( "1400/280 = (1 + r) ^ 5.<BR>\n" );
document.write( "take the fifth root of both sides of this equation to get:<BR>\n" );
document.write( "(1400/280)^(1/5) = 1 + r.<BR>\n" );
document.write( "subtract 1 from both sides of this equation to get:<BR>\n" );
document.write( "(1400/280)^(1/5) - 1 = r<BR>\n" );
document.write( "solve for r to get:<BR>\n" );
document.write( "r = .379729661<BR>\n" );
document.write( "that's the rate of growth per hour.\r<BR>\n" );
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document.write( "replace r in the original equation and solve for f to get:<BR>\n" );
document.write( "f = 280  * (1 + .379729661) ^ 5 = 1400.\r<BR>\n" );
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document.write( "the rate is good.<BR>\n" );
document.write( "after 4 hours, the population will be:<BR>\n" );
document.write( "f = 280 * (1 + .379729661) ^ 4 = 1014.691529.\r<BR>\n" );
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document.write( "1014.691529 is your solution to question a.<BR>\n" );
document.write( "i'm not sure how they want you to round this so i left it as is.\r<BR>\n" );
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document.write( "to find how long it takes to grow to 2630, the formula becomes:\r<BR>\n" );
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document.write( "2630 = 280 * (1 + .379729661) ^ n<BR>\n" );
document.write( "divide both sides of the equation to 280 and simplify the right side of the equation to get:<BR>\n" );
document.write( "2630/280 = 1.379729661 ^ n<BR>\n" );
document.write( "take the log of both sides of the equation to get:<BR>\n" );
document.write( "log(2630/280) = log(1.379729661 ^ n)<BR>\n" );
document.write( "since log(x^n) = n*log(x), this becomes:<BR>\n" );
document.write( "log(2630/280) = n * log(1.379729661).<BR>\n" );
document.write( "divide both sides of the equation by log(1.379729661) to get:<BR>\n" );
document.write( "log(2630/280) / log(1.379729661) = n<BR>\n" );
document.write( "solve for n to get:<BR>\n" );
document.write( "n = 6.958794448.\r<BR>\n" );
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document.write( "confirm by replacing n in the original equation and solving for f to get:<BR>\n" );
document.write( "f = 280 * 1.379729661 ^ 6.958794448 = 2630.<BR>\n" );
document.write( "round to 2 decimal places to get n = 6.96\r<BR>\n" );
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document.write( "the equation 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/2021/120912.jpg\" >\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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