document.write( "Question 79412: suppose a population doubles every 20 years. Then by what percentage does it increase each year? \n" ); document.write( "
Algebra.Com's Answer #56995 by Edwin McCravy(20060)\"\" \"About 
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\r\n" );
document.write( "suppose a population doubles every 20 years. \r\n" );
document.write( "Then by what percentage does it increase \r\n" );
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document.write( "This is the same problem and has the same \r\n" );
document.write( "answer as this one:\r\n" );
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document.write( "$1000 deposited in a savings account compounded \r\n" );
document.write( "annually grows to $2000 in 20 years.  What is \r\n" );
document.write( "the rate of interest?\r\n" );
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document.write( "\"A+=+P%281+%2B+r%2Fn%29%5E%28nt%29\"\r\n" );
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document.write( "with A = 2000, P = 1000, \r\n" );
document.write( "r unknown, n = 1 (compounded 1 time a yaer),\r\n" );
document.write( "n = 20.\r\n" );
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document.write( "\"2000+=+1000%281+%2B+r%2F1%29%5E%281%2A20%29\"\r\n" );
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document.write( "Divide both sides by 1000\r\n" );
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document.write( "\"2+=+%281+%2B+r%29%5E20\"\r\n" );
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document.write( "Take the twentieth root of both sides\r\n" );
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document.write( "\"root%2820%2C2%29+=+1+%2B+r\"\r\n" );
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document.write( "\"root%2820%2C2%29-1+=+r\"\r\n" );
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document.write( "\"1.035264924-1=r\"\r\n" );
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document.write( "\".035264924=r\"\r\n" );
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document.write( "So the yearly rate is slightly over 3.5%. \r\n" );
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document.write( "Or, if your teacher doesn't like to pretend it's money, then \r\n" );
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document.write( "Let x be the yearly percentage expressed as a decimal.\r\n" );
document.write( "Suppose the population at the beginning of the first year is P.\r\n" );
document.write( "Then:\r\n" );
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document.write( "At the end of the 1st year it is P + Px = P(1 + x)\r\n" );
document.write( "At the end of the 2nd year it is P(1 + x) + xP(1 + x) = P(1 + x)2\r\n" );
document.write( "At the end of the 3rd year it is P(1 + x)2 + xP(1 + x) = P(1 + x)3\r\n" );
document.write( "...\r\n" );
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document.write( "At the end of the 20th year it is P(1 + x)20\r\n" );
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document.write( "Since it has doubled at the end of 20 years we set that equal to 2P\r\n" );
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document.write( "     P(1 + x)20 = 2P\r\n" );
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document.write( "Divide through by P\r\n" );
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document.write( "      (1 + x)20 = 2\r\n" );
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document.write( "Take thwentith roots of both sides\r\n" );
document.write( "                    \r\n" );
document.write( "         1 + x = \"root%2820%2C2%29\" \r\n" );
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document.write( "             x = \"root%2820%2C2%29\" - 1\r\n" );
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document.write( "             x = 1.035264924 - 1\r\n" );
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document.write( "             x = .035264924 or slightly over 3.5%\r\n" );
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document.write( "Edwin

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