document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277891>Question 277891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The population of a small town has been gradually decreasing over the past number of years.  The equation: A = 1236(.97)t, where t is time in years, models the decrease in the town’s population.\r<BR>\n" );
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document.write( "In how many years will it take for the population to reach 846 people? (Round to the nearest hundredth of a year.)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #202309 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=202309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your formula should read:\r<BR>\n" );
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document.write( "A = 1236 * (.97)^T where:\r<BR>\n" );
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document.write( "A is the future population.<BR>\n" );
document.write( "T is the time in years.\r<BR>\n" );
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document.write( "The ^ indicates exponentiation.\r<BR>\n" );
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document.write( "According to the equation model, the initial population is 1236.\r<BR>\n" );
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document.write( "The general form of the equation is:\r<BR>\n" );
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document.write( "F = P * (1+G)^T where:\r<BR>\n" );
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document.write( "F = the future population<BR>\n" );
document.write( "P = the present population<BR>\n" );
document.write( "G = the annual growth rate<BR>\n" );
document.write( "T = the number of years\r<BR>\n" );
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document.write( "In your equation:\r<BR>\n" );
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document.write( "F = A<BR>\n" );
document.write( "P = 1236<BR>\n" );
document.write( "G = -.03 because 1 - .03 = .97<BR>\n" );
document.write( "T = number of years\r<BR>\n" );
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document.write( "Your equation becomes:\r<BR>\n" );
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document.write( "A = 1236 * .97^T\r<BR>\n" );
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document.write( "Since A is the future population, then A becomes 846.\r<BR>\n" );
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document.write( "Your equation becomes:\r<BR>\n" );
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document.write( "846 = 1236 * .97^T\r<BR>\n" );
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document.write( "Divide both sides of this equation by 1236 to get:\r<BR>\n" );
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document.write( "846/1236 = .97^T\r<BR>\n" );
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document.write( "Take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(846/1236) = log(.97^T)\r<BR>\n" );
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document.write( "Since log(a^b) = b*log(a), your equation becomes:\r<BR>\n" );
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document.write( "log(846/1236) = T*log(.97)\r<BR>\n" );
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document.write( "Divide both sides of this equation by log(.97) to get:\r<BR>\n" );
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document.write( "log(846/1236)/log(.97) = T which is the same as:\r<BR>\n" );
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document.write( "T = log(846/1236)/log(.97)\r<BR>\n" );
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document.write( "use your calculator to find the logs which make your equation equal to:\r<BR>\n" );
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document.write( "T = -.164648108 / -.013228266 = 12.44668886\r<BR>\n" );
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document.write( "The future population will be 846 in 12.4466886 years.\r<BR>\n" );
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document.write( "1236 * .97^(12.44668886) = 846\r<BR>\n" );
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document.write( "Round that to the nearest hundredth of a year and you get:\r<BR>\n" );
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document.write( "T = 12.45\r<BR>\n" );
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