document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153464>Question 1153464</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bacteria culture starts with 560 bacteria and grows at a rate proportional to its size. After 2 hours there will be 1120 bacteria.\r<BR>\n" );
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document.write( "(a) Express the population P after t hours as a function of t.\r<BR>\n" );
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document.write( "P(t)=____\r<BR>\n" );
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document.write( "(b) What will be the population after 8 hours?\r<BR>\n" );
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document.write( "___ bacteria\r<BR>\n" );
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document.write( "(c) How long will it take for the population to reach 2490? Give your answer accurate to at least 2 decimal places.\r<BR>\n" );
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document.write( "___ hours </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #775680 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=775680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1120/560 = 2<BR>\n" );
document.write( "the formula to use for this problem is:<BR>\n" );
document.write( "p(t) = 560 * (1 + r) ^ t<BR>\n" );
document.write( "p(t) becomes 1120 and t becomes 2 to get:<BR>\n" );
document.write( "1120 = 560 * (1 + r) ^ 2<BR>\n" );
document.write( "divide both sides by 560 to get:<BR>\n" );
document.write( "1120 / 560 = (1 + r) ^ 2<BR>\n" );
document.write( "take the square root of both sides to get:<BR>\n" );
document.write( "(1120 / 560) ^ (1/2) = 1 + r<BR>\n" );
document.write( "solve for 1 + r to get:<BR>\n" );
document.write( "1 + r = 1.414215362<BR>\n" );
document.write( "that's your hourly growth factor.<BR>\n" );
document.write( "in 2 hours, you get 560 * 1.414215362 ^ 2 = 1120<BR>\n" );
document.write( "in 8 hours, you get 560 * 1.414215362 ^ 8 = 8960<BR>\n" );
document.write( "to find how long for the population to becomes 2490, you would do the following.<BR>\n" );
document.write( "formula is p(t) = 560 * 1.414215362 ^ t<BR>\n" );
document.write( "when p(t) is 2490, the formula becomes:<BR>\n" );
document.write( "2490 = 560 * 1.414215362 ^ t<BR>\n" );
document.write( "divide both sides by 560 to get<BR>\n" );
document.write( "2490 / 560 = 1.414215362 ^ t<BR>\n" );
document.write( "take the log of both sides of the equation to get:<BR>\n" );
document.write( "log(2490 / 560) = log(1.414215362 ^ t)<BR>\n" );
document.write( "since log(x^a) = a * log(x), your formula becomes:<BR>\n" );
document.write( "log(2490 / 560) = t * log(1.414215362<BR>\n" );
document.write( "divide both sides by log(1.414215362) to get:<BR>\n" );
document.write( "log(2490 / 560) / log(1.414215362) = t<BR>\n" );
document.write( "solve for t to get:<BR>\n" );
document.write( "t = 4.30529402.<BR>\n" );
document.write( "confirm by replacing t with that in the original equation to get:<BR>\n" );
document.write( "p(t) = 560 * 1.414215362 ^ 4.30529402 = 2490<BR>\n" );
document.write( "this confirms the solution is correct.<BR>\n" );
document.write( "you can graph the equation as y = 560 * 1.414215362 ^ x<BR>\n" );
document.write( "it would look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/030402.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the graphing software rounds the answer to 3 decimal places.<BR>\n" );
document.write( "4.3052.... becomes 4.305.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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