document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167428>Question 1167428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> City A is growing at a rate of 6.5%/year and starts with a population of 100,000. City B has a growth rate of 4.58%/year and starts with a population of 250,000.<BR>\n" );
document.write( "(a) Model the population of City A.<BR>\n" );
document.write( "(b) Model the population of City B.<BR>\n" );
document.write( "(c) What will the population of City A be in 10 years? Estimate using two decimal places.<BR>\n" );
document.write( "(d) When will the population of City B reach 300,000? Estimate using two decimal places.<BR>\n" );
document.write( "(e)When will the population of City A equal the population of City B? Estimate using<BR>\n" );
document.write( "two decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792007 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=792007\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula to use if f = p * (1 + r) ^ n\r<BR>\n" );
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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 year = percent growth rate per year / 100<BR>\n" );
document.write( "n is the number of years.\r<BR>\n" );
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document.write( "for city A, the formula becomes f = 100,000 * (1 + .065) ^ n\r<BR>\n" );
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document.write( "for city B, the formula becomes f = 250,000 * (1 + .0458) ^ n\r<BR>\n" );
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document.write( "the population for city A in 10 years would be modeled as:\r<BR>\n" );
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document.write( "f = 100,000 * (1 + .065) ^ 10\r<BR>\n" );
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document.write( "solve for f to get f = 187,713.7465.\r<BR>\n" );
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document.write( "the population for city B reaching 300,000 would be modeled as:\r<BR>\n" );
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document.write( "300,000 = 250,000 * (1 + .0458) ^ n\r<BR>\n" );
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document.write( "divide both sides of this formula by 250,000 to get:\r<BR>\n" );
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document.write( "300,000 / 250,000 = (1 + .0458) ^ n\r<BR>\n" );
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document.write( "simplify a little and take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(300/250) = log((1 + .0458) ^ n)\r<BR>\n" );
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document.write( "since log((1 + .0458) ^ n) = n * log(1 + .0458), then the equation becomes:\r<BR>\n" );
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document.write( "log(300/250) = n * log(1 + .0458)\r<BR>\n" );
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document.write( "solve for n to get:\r<BR>\n" );
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document.write( "n = log(300/250) / log(1.0458) = 4.071300423 years.\r<BR>\n" );
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document.write( "confirm by replacing n with that in the original equation to get:\r<BR>\n" );
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document.write( "300,000 = 250,000 * (1 + .0458) ^ n becomes:<BR>\n" );
document.write( "300,000 = 250,000 * (1 + .0458) ^ 4.071300423 which becomes:<BR>\n" );
document.write( "300,000 = 300,000\r<BR>\n" );
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document.write( "this confirms that the solution is that a population of 250,000 will become 300,000 in 4.071300423 years when the growth rate if 4.58% per year.\r<BR>\n" );
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document.write( "the population of city A will become equal to the population of city B when 100,000 * (1 + .065) ^ n is equal to 250,000 * (1 + .0458) ^ n\r<BR>\n" );
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document.write( "this is because f1 = 100,000 * (1 + .065) ^ n for city A and f2 = 250,000 * (1 + .0458) ^ n for city B.\r<BR>\n" );
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document.write( "f1 is the future value for city A.<BR>\n" );
document.write( "f2 is the future value for city B.\r<BR>\n" );
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document.write( "for f1 to be equal to f2, then 100,000 * (1 + .065) ^ n must be equal to 250,000 * (1 + .0458) ^ n\r<BR>\n" );
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document.write( "the formula to solve is therefore:\r<BR>\n" );
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document.write( "100,000 * (1 + .065) ^ n = 250,000 * (1 + .0458) ^ n\r<BR>\n" );
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document.write( "divide both sides of this equation by 250,000 and divide both sides of this equation by (1 + .065) ^ n to get:\r<BR>\n" );
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document.write( "100,000 / 250,000 = (1 + .0458) ^ n / (1 + .065) ^ n\r<BR>\n" );
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document.write( "simplify a little to get:\r<BR>\n" );
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document.write( "100/250 = 1.0458 ^ n / 1.065 ^ n\r<BR>\n" );
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document.write( "since 1.0458 ^ n / 1.065 ^ n is equal to (1.0458 / 1.065) ^ n, then the formula becomes:\r<BR>\n" );
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document.write( "100 / 250 = (1.0458 / 1.065) ^ n\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 (100 / 250) = n * log(1.0458 / 1.065)\r<BR>\n" );
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document.write( "solve for n to get:\r<BR>\n" );
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document.write( "n = log(100 / 250) / log(1.0458 / 1.065)\r<BR>\n" );
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document.write( "this gets you:\r<BR>\n" );
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document.write( "n = 50.36596703 years.\r<BR>\n" );
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document.write( "the formula for city A becomes f = 100,000 * 1.065 ^ 50.36596703.<BR>\n" );
document.write( "the formula for city B becomes f = 250,000 * 1.0458 ^ 50.36596703.\r<BR>\n" );
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document.write( "solve for f in both formulas to get:\r<BR>\n" );
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document.write( "the population for city A will become 2,385,005.821 in 50.36596703 years.<BR>\n" );
document.write( "the population for city B will become 2,385,005.821 in 50.36596703 years.\r<BR>\n" );
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document.write( "they're the same in 50.36596703 years.\r<BR>\n" );
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document.write( "answer to your questions are:\r<BR>\n" );
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document.write( "(a) Model the population of City A.\r<BR>\n" );
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document.write( "f = 100,000 * 1.065 ^ n\r<BR>\n" );
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document.write( "(b) Model the population of City B.\r<BR>\n" );
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document.write( "f = 250,000 * 1.0458 ^ n\r<BR>\n" );
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document.write( "(c) What will the population of City A be in 10 years? Estimate using two decimal places.\r<BR>\n" );
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document.write( "population of city A will be 187,713.75 in 10 years.\r<BR>\n" );
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document.write( "(d) When will the population of City B reach 300,000? Estimate using two decimal places.\r<BR>\n" );
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document.write( "the population of city B will reach 300,000 in 4.07 years.\r<BR>\n" );
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document.write( "(e)When will the population of City A equal the population of City B? \r<BR>\n" );
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document.write( "the population of city A will be equal to the population of city B in 50.37 years.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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