document.write( "Question 62494: Use a calculator to help solve each problem.
\n" ); document.write( "This regarding radioactive decay. Can anyone give me a hand? Is the formula the same as carbon-14 dating?\r
\n" ); document.write( "\n" ); document.write( "In 2 years, 20% of radioactive element
\n" ); document.write( "decays. Find its half life.\r
\n" ); document.write( "\n" ); document.write( "Thanks for your help! \n" ); document.write( "
Algebra.Com's Answer #43245 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
Use a calculator to help solve each problem.\r\n" );
document.write( "This regarding radioactive decay. \r\n" );
document.write( "Can anyone give me a hand? Is the formula \r\n" );
document.write( "the same as carbon-14 dating? \r\n" );
document.write( "In 2 years, 20% of radioactive element\r\n" );
document.write( "decays. Find its half life. \r\n" );
document.write( "Thanks for your help!\r\n" );
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document.write( "The problem can be stated equivalently this way:\r\n" );
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document.write( "After 2 years, 80% of a quantity of radioactive \r\n" );
document.write( "element remains.  After how many years will only \r\n" );
document.write( "50% of it remain? \r\n" );
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document.write( "The formula for all exponential growth or decay is\r\n" );
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document.write( "A = Pert\r\n" );
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document.write( "It is a growth when r is positive and a decay when \r\n" );
document.write( "r is negative. P represents the original amount \r\n" );
document.write( "and A represents the final amount after t units of \r\n" );
document.write( "time.\r\n" );
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document.write( "Suppose we begin with P units of this radioactive \r\n" );
document.write( "element.  Then when t = 2 years, 20% of P decays, \r\n" );
document.write( "leaving 80% of it, or .8P units remain.\r\n" );
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document.write( "So we substitute A = .8P, and t = 2.\r\n" );
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document.write( ".8P = Per(2)\r\n" );
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document.write( "Divide both sides by P and write r(2) as 2r\r\n" );
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document.write( ".8 = e2r\r\n" );
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document.write( "Use the rule that says any equation of the\r\n" );
document.write( "form A = eB can be rewritten B = ln(A). \r\n" );
document.write( "We may rewrite the above equation as\r\n" );
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document.write( "2r = ln(.8)\r\n" );
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document.write( "r = ln(.8)/2 = -.1115717757\r\n" );
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document.write( "Since this is a decay, we expected r to be \r\n" );
document.write( "negative.  Now we substitute this value of r\r\n" );
document.write( "in the equartion\r\n" );
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document.write( "A = Pert\r\n" );
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document.write( "A = Pe-.1115717757t\r\n" );
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document.write( "Now we wish to know its half life, or how\r\n" );
document.write( "many years it will take P units of the \r\n" );
document.write( "radioactive subatance to decay to only 50% \r\n" );
document.write( "of P units or .5P units.\r\n" );
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document.write( "Su we substitute .5P for A:\r\n" );
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document.write( ".5P = Pe-.1115717757t\r\n" );
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document.write( "Divide both sides by P\r\n" );
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document.write( ".5 = e-.1115717757t\r\n" );
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document.write( "Rewrite this equation as\r\n" );
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document.write( "-.1115717757t = ln(.5)\r\n" );
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document.write( "Divide both sides by -.1115717757\r\n" );
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document.write( "            t = ln(.5)/(-.1115717757)\r\n" );
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document.write( "            t = 6.212567439 years\r\n" );
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document.write( "So its half life is about 6.2 years.\r\n" );
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document.write( "Edwin
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