document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171138>Question 1171138</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	 <BR>\n" );
document.write( "In​ 2012, the population of a city was 5.21 million. The exponential growth rate was ​2.86% per year.\r<BR>\n" );
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document.write( "a) Find the exponential growth function.<BR>\n" );
document.write( "​b) Estimate the population of the city in 2018.<BR>\n" );
document.write( "​c) When will the population of the city be 9 ​million?<BR>\n" );
document.write( "​d) Find the doubling time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #796036 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=796036\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> general form of exponential formula is y = a * b^x<BR>\n" );
document.write( "a is a constant.<BR>\n" );
document.write( "b is the base<BR>\n" );
document.write( "x is the exponent.\r<BR>\n" );
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document.write( "in this problem, a = 5.21 million and b = 1 + 2.86%/100 = 1.0286.\r<BR>\n" );
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document.write( "in 2012, the population was 5.21 million.<BR>\n" );
document.write( "in 2018, the population will be 5.21 * 1.0286 ^ (2018 - 2012) = 5.21 * 1.0286 ^ 6 = 6.170450081 million.\r<BR>\n" );
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document.write( "to find when the population will be 9 million, your equation becomes:<BR>\n" );
document.write( "9 = 5.21 * 1.0286 ^ x<BR>\n" );
document.write( "divide both sides of this formula by 5.21 to get:<BR>\n" );
document.write( "9/5.21 = 1.0286 ^ x<BR>\n" );
document.write( "take the log of both sides of thie equation to get:<BR>\n" );
document.write( "log(9/5.21) = log(1.0286^x)<BR>\n" );
document.write( "since log(1.0286^x) = x * log(1.0286), the equation becomes:<BR>\n" );
document.write( "log(9/5.21) = x * log(1.0286)<BR>\n" );
document.write( "divide both sides of this equation by log(1.0286) to get:<BR>\n" );
document.write( "log(9/5.21) / log(1.0286) = x<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 19.38548963.<BR>\n" );
document.write( "confirm by replace x in the original equation to get:<BR>\n" );
document.write( "9 = 5.21 * 1.0286 ^ 19.38548963 becomes 9 = 9.<BR>\n" );
document.write( "this confirms the value of x is good.<BR>\n" );
document.write( "the population will grow to 9 million in 19.38548963 years.\r<BR>\n" );
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document.write( "to find when the population will double, the equation becomes:<BR>\n" );
document.write( "2 = 1 * 1.0286 ^ x<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "2 = 1.0286 ^ x<BR>\n" );
document.write( "take the log of both sides of this equation to get:<BR>\n" );
document.write( "log(2) = log(1.0286 ^ x)<BR>\n" );
document.write( "since log(1.0286 ^ x) = x * log(1.0286), this equation becomes:<BR>\n" );
document.write( "log(2) = x * log(1.0286)<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = log(2) / log(1.0286) = 24.5808602.<BR>\n" );
document.write( "to confirm this is true, replace x in the original equation to get:<BR>\n" );
document.write( "2 = 1.0286 ^ 24.5808602 which becomes 2 = 2.<BR>\n" );
document.write( "this confirms the value of x is true.<BR>\n" );
document.write( "the population will double in 24.5808602 years.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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