document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1025348>Question 1025348</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The population of insects on an island has grown by 50% over an 5 month period. For simplicity, we<BR>\n" );
document.write( "will assume that the percentage growth per month has been constant, and compunded each month.<BR>\n" );
document.write( "(After all, the increase in the population should depend on the current population, not what it was<BR>\n" );
document.write( "a month, or 5 months, ago.)<BR>\n" );
document.write( "(a) Without calculation, do you excpect the monthly percentage increase to be 10%, more than<BR>\n" );
document.write( "10%, or less than 10%?<BR>\n" );
document.write( "(b) Find the monthly percentage increase, both exactly, and also to 3 significant figures.<BR>\n" );
document.write( "(c) By a process of guessing and checking, find out, to the nearest half month†<BR>\n" );
document.write( ", how long it will<BR>\n" );
document.write( "take for the population of insects to double. It may be useful to note that if a quantity doubles,<BR>\n" );
document.write( "it has increased by 100%.<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 #640704 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=640704\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> with compounding you would expect the monthly percent increase to be less than 10%.\r<BR>\n" );
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document.write( "it would be exactly 10% if it was straight line growth.\r<BR>\n" );
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document.write( "with compounding 10% a month for 5 months would be greater than 50% growth.\r<BR>\n" );
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document.write( "you would therefore expect the monthly growth rate to be less than 10% when you use compounding.\r<BR>\n" );
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document.write( "to find the monthly percent increase, use the following general formula.\r<BR>\n" );
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document.write( "f = p * (1 + r) ^ n\r<BR>\n" );
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document.write( "r is the interest rate per month.<BR>\n" );
document.write( "n is the number of months.\r<BR>\n" );
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document.write( "p is equal to 1<BR>\n" );
document.write( "f is equal to 1.5 because the number of flies increased by 50% in 5 months.\r<BR>\n" );
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document.write( "formula becomes 1.5 = 1 * (1 + r) ^ 5\r<BR>\n" );
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document.write( "simplify to get 1.5 = (1 + r) ^ 5.\r<BR>\n" );
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document.write( "raise each side of the equation to the power of 1/5 to get:\r<BR>\n" );
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document.write( "1.5 ^ (1/5) = ((1 + r) ^ 5) ^ (1/5\r<BR>\n" );
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document.write( "you will get 1.5 ^ (1/5) = 1 + r\r<BR>\n" );
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document.write( "subtract 1 from both sides of this equation to get:\r<BR>\n" );
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document.write( "1.5 ^ (1/5) - 1 = r\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = .0844717712\r<BR>\n" );
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document.write( "round to 3 significant digits to get r = .0845.\r<BR>\n" );
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document.write( "you want to know how long it will take for the population of insects to double.\r<BR>\n" );
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document.write( "since f = p * (1 + r) ^ n, and since:\r<BR>\n" );
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document.write( "f = 2<BR>\n" );
document.write( "p = 1<BR>\n" );
document.write( "1 + r = 1.0844717712, then:\r<BR>\n" );
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document.write( "formula becomes 2 = 1.0844717712 ^ n\r<BR>\n" );
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document.write( "you want to solve for n.\r<BR>\n" );
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document.write( "you can use logs or you can do guess and check.\r<BR>\n" );
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document.write( "we'll do guess and check, since that's what they asked you to do.\r<BR>\n" );
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document.write( "you know n is going to be greater than 5, because f = 1.5 when n = 5.\r<BR>\n" );
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document.write( "try n = 10.\r<BR>\n" );
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document.write( "store 1.0844717712 into memory of your calculator to make subsequent calculations easy.\r<BR>\n" );
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document.write( "1.0844717712 ^ 10 = 2.25\r<BR>\n" );
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document.write( "make n = 8.\r<BR>\n" );
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document.write( "you get 1.0844717712 ^ 8 = 1.91.....\r<BR>\n" );
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document.write( "make n = 9.\r<BR>\n" );
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document.write( "you get 1.0844717712 ^ 9 = 2.074.....\r<BR>\n" );
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document.write( "the number of insects will double between the 8th month and the 9th month.\r<BR>\n" );
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document.write( "you can try n = 8.5 to get 1.0844717712 ^ 8.5 = 1.992....\r<BR>\n" );
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document.write( "you can try n = 8.6 to get 1.0844717712 ^ 8.6 = 2.008.\r<BR>\n" );
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document.write( "the number of insects will double sometime between the 8.5th and 8.6th month.\r<BR>\n" );
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document.write( "to the nearest half month, it would take 8.5 months for the population to double.\r<BR>\n" );
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document.write( "if i were to solve exactly, i would use logs.\r<BR>\n" );
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document.write( "i would start with 2 = 1.0844717712 ^ n\r<BR>\n" );
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document.write( "i would take the log of both sides of the equation to get:\r<BR>\n" );
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document.write( "log(2) = log(1.0844717712 ^ n)\r<BR>\n" );
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document.write( "by the laws of logarithms, this would becomes log(2) = n * log(1.0844717712).\r<BR>\n" );
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document.write( "i would solve for n to get n = log(2) / log(1.0844717712) = 8.547556457 months.\r<BR>\n" );
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document.write( "round that to the nearest 10th and you get 8.5 months.\r<BR>\n" );
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document.write( "the guess and check method and the log method agree with each other.\r<BR>\n" );
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document.write( "if you didn't take logs yet, don't worry about it.\r<BR>\n" );
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document.write( "you will, eventually.\r<BR>\n" );
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document.write( "a note on significant figures, which i determined to mean significant digits.\r<BR>\n" );
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document.write( "that's different than rounding to s set number of decimal places.\r<BR>\n" );
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document.write( ".0844717712 rounded to 3 decimal places would give you .084\r<BR>\n" );
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document.write( "1.0844717712 round to 3 significant digits would give you 1.0845.\r<BR>\n" );
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document.write( "this is based on the following reference which discusses significant digits.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/rounding2.htm\" target = \"_blank\">http://www.purplemath.com/modules/rounding2.htm</a>\r<BR>\n" );
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document.write( "the bad news is that it can get a little complex to figure it out.\r<BR>\n" );
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document.write( "the good news is that you won't have to do it that often.\r<BR>\n" );
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document.write( "most of the time you will just round to the number of decimal places desired.\r<BR>\n" );
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document.write( "nearest tenth, nearest one hundredth, etc.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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