document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198714>Question 1198714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In​ 2012, the population of a city was 5.62 million. The exponential growth rate was ​2.41% per year.<BR>\n" );
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 9million?<BR>\n" );
document.write( "​d) Find the doubling time.<BR>\n" );
document.write( "I already asked this question but I need more Clarification  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833178 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=833178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula to use is f = p * (1+r) ^ n<BR>\n" );
document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "r is the growth rate per time period<BR>\n" );
document.write( "(1+r) is the growth factor per time period<BR>\n" );
document.write( "n is the number of time periods<BR>\n" );
document.write( "p is equal to 5.62<BR>\n" );
document.write( "r is equal to 2.41% / 100 = .0241<BR>\n" );
document.write( "1+r is equal to 1.0241<BR>\n" );
document.write( "n is equal to 2018 minus 2012 = 6<BR>\n" );
document.write( "formula becomes f = 5.62 * (1+r) ^ 6<BR>\n" );
document.write( "solve for f to get f = 6.483216317.<BR>\n" );
document.write( "answer to (a) ia f = p * (1+r) ^ n<BR>\n" );
document.write( "answer to (b) is 6.483216317<BR>\n" );
document.write( "answers to c and d are shown below.<BR>\n" );
document.write( "c) When will the population of the city be 9 million?<BR>\n" );
document.write( "to find this, replace f with 9 in the equation to solve for b and replace 6 with n to get:<BR>\n" );
document.write( "9 = 5.62 * 1.0241 ^ n<BR>\n" );
document.write( "divide both sides of the equation by 5.62 to get:<BR>\n" );
document.write( "9/5.62 = 1.0241 ^ n<BR>\n" );
document.write( "take the natural log of both sides of the equation to get:<BR>\n" );
document.write( "ln(9/5.62) = ln(1.0241 ^ n)<BR>\n" );
document.write( "since ln(1.0241 ^ n) = n * ln(.0241), the equation becomes:<BR>\n" );
document.write( "ln(9/5.62) = n * ln(1.0241)<BR>\n" );
document.write( "divide both sides of the equation by ln(1.0241) to get:<BR>\n" );
document.write( "ln(9/5.62) / ln(1.0241) = n<BR>\n" );
document.write( "solve for n to get:<BR>\n" );
document.write( "n = 19.77363701<BR>\n" );
document.write( "to confirm, replace n in the equation with that and solve for f to get:<BR>\n" );
document.write( "f = 5.62 * 1.0241 ^ 19.77363701 = 9, confirming the value of n is correct.<BR>\n" );
document.write( "since your base year is 2012, add 19.7763701 to that to get 2031.773637.<BR>\n" );
document.write( "that should be in the year sometime in the year 2031.<BR>\n" );
document.write( "​d) Find the doubling time.<BR>\n" );
document.write( "to find the doubling time, multiply 5.62 * 2 to get 11.24<BR>\n" );
document.write( "equation becomes 11.24 = 5.62 * 1.0241 ^ n<BR>\n" );
document.write( "divide both sides of the equation by 5.62 to get:<BR>\n" );
document.write( "2 = 1.0241 ^ n<BR>\n" );
document.write( "take natural log of both sides of the equation to get:<BR>\n" );
document.write( "ln(2) = ln(1.0241^n)<BR>\n" );
document.write( "since ln(1.0241^n) is equal to n * ln(1.0241), equation becomes:<BR>\n" );
document.write( "ln(2) = n * ln(1.0241)<BR>\n" );
document.write( "divide both sides of the equation by ln(1.0241) and solve for n to get:<BR>\n" );
document.write( "n = ln(2) / ln(1.0241) = 29.10649184<BR>\n" );
document.write( "that's the number of years it will take for 5.62 to double.<BR>\n" );
document.write( "confirm by replacing n with that in the equation and solving for f to get:<BR>\n" );
document.write( "f = 5.62 * 1.0241 ^ 29.20619184 = 11.24, confirming the value of n is good.<BR>\n" );
document.write( "your solution is that it will take 29.20619184 years for the population to double.\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/2022/122203.jpg\".\r<BR>\n" );
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document.write( "i'm not sure if this is going to help you, but i thought i would give it a try.<BR>\n" );
document.write( "let me know if you have any questions.<BR>\n" );
document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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