document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188359>Question 1188359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When the population growth of a certain city was first studied, the population was 22,000. It was found that the population P grows with respect to time t in years by the formula P=22,000(10exponential of 0.163t)\r<BR>\n" );
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document.write( "How long will it take for the city to double its population?<BR>\n" );
document.write( "How long will it take for the population of the city to triple? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819448 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=819448\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the growth formula is either:\r<BR>\n" );
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document.write( "f = p * (1 + r) ^ n\r<BR>\n" );
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document.write( "or:\r<BR>\n" );
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document.write( "f = p * e ^ (r * t)\r<BR>\n" );
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document.write( "i don't understand p = 22,000 * 10 ^ .163t.\r<BR>\n" );
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document.write( "i suspect you want f = p * e ^ (r * t)\r<BR>\n" );
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document.write( "under that aswsumption, .....\r<BR>\n" );
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document.write( "for the population to double, the formula becomes:\r<BR>\n" );
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document.write( "44,000 = 22,000 * e ^ (.163 * t)\r<BR>\n" );
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document.write( "divide both sides of the formula by 22,000 to get:\r<BR>\n" );
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document.write( "2 = e ^ (.163 * t)\r<BR>\n" );
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document.write( "take the natural log of both sides of the formula to get:\r<BR>\n" );
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document.write( "ln(2) = ln(e ^ (.163 * t)) which becomes:\r<BR>\n" );
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document.write( "ln(2) = .163 * t * ln(e) which becomes:\r<BR>\n" );
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document.write( "ln(2) = .163 * t\r<BR>\n" );
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document.write( "solve for t to get:\r<BR>\n" );
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document.write( "t = ln(2) / .163 = 4.252436691\r<BR>\n" );
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document.write( "confirm by replacing t in the original requation of f = 22,000 * e ^ (.163 * t) with 4.252436691 to get:\r<BR>\n" );
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document.write( "f = 22,000 * e ^ (.163 * 4.252436691).\r<BR>\n" );
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document.write( "solve for f to get:\r<BR>\n" );
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document.write( "f = 44000.\r<BR>\n" );
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document.write( "to triple the growth, you'll go through the same steps from:\r<BR>\n" );
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document.write( "66,000 = 22,000 * e ^ (.163 * t)\r<BR>\n" );
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document.write( "and wind up with:\r<BR>\n" );
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document.write( "ln(3) = .163 * t\r<BR>\n" );
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document.write( "solve for t to get:\r<BR>\n" );
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document.write( "t = ln(3) / .163 = 6.739952691\r<BR>\n" );
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document.write( "confirm by replacing t in the original equation of f = 22,000 * e ^ (.163 * t) with that to get:\r<BR>\n" );
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document.write( "f = 22,000 * e ^ (.163 * 6.739952691) \r<BR>\n" );
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document.write( "solve for f to get:\r<BR>\n" );
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document.write( "f = 66000.\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/2021/120209.jpg\" >\r<BR>\n" );
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document.write( "the coordinate points are in (x,y) format.\r<BR>\n" );
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document.write( "x in the graph equation is the same as t in the algebra equation.\r<BR>\n" );
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document.write( "y in the graph equation is the same as f in the algebra equation.\r<BR>\n" );
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document.write( "what this means is:\r<BR>\n" );
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document.write( "f = 22000 * e ^ (.163 * t) and y = 22000 * e ^ (.163 * x) are equivalent equations and will give you the same results.\r<BR>\n" );
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document.write( "f and y represent the future value of the population.<BR>\n" );
document.write( "t and x represent the number of years.\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 </SPAN>\n" );
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