document.write( "Question 1195759: The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon had lost 60.4% of their carbon-14. How old were the bones at the time they were discovered?
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Algebra.Com's Answer #828275 by ikleyn(52878) $\"\"$ $\"About$

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\n" ); document.write( "The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones
\n" ); document.write( "from a mastodon had lost 60.4% of their carbon-14. How old were the bones at the time they were discovered?
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\n" ); document.write( "\n" ); document.write( "On solving similar problem, related to radioactive decay, see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Radioactive decay problems \r
\n" ); document.write( "\n" ); document.write( "    - Carbon dating problems \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Learn the subject from there.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Logarithms\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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\n" ); document.write( "\n" ); document.write( "            The solution by @josgaithmetic in his post is INCORRECT from its first line
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\n" ); document.write( "\n" ); document.write( "            In my post, I will bring a correct solution, and then at the end, will explain,
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document.write( "Since you are given half-life period, you do not need to write the exponential\r\n" );
document.write( "decay function in the (ekt)-form and then to determine the parameter \"k\" of this form.\r\n" );
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document.write( "Write exponential decay function for the remaining mass using base 1/2 in the form\r\n" );
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document.write( "    M(t) = .    (1)\r\n" );
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document.write( "It is the STANDARD exponential decay form function, when the half-life is given.\r\n" );
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document.write( "Next, in your problem, the lost mass of the carbon-14 is 60.4%.\r\n" );
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document.write( "It means that remaining mass of carbon-14 is 100%-60.4% = 39.6%.\r\n" );
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document.write( "THEREFORE, the equation (1) for the remaining mass takes the form\r\n" );
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document.write( "    0.396 = ,\r\n" );
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document.write( "or, which is the same,\r\n" );
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document.write( "    0.396 = 0.5^(t/5750).    (2)\r\n" );
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document.write( "Take logarithm base 10 of both sides of (2).  You will get\r\n" );
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document.write( "    log(0.396) = \r\n" );
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document.write( "which implies\r\n" );
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document.write( "    t =  = 7684 years, or, rounding to closest one hundred years,\r\n" );
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document.write( "as usually people do in such calculations, we get the \r\n" );
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document.write( "ANSWER.  The bones are 7700 years old.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The solution by @josgarithmetic has two errors.\r
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\n" ); document.write( "\n" ); document.write( "First error is that he starts from (ekt)-form of the exponential decay function,
\n" ); document.write( "having given data about the half life period.\r
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\n" ); document.write( "\n" ); document.write( "In such cases, the normal procedure is to use the base 1/2 exponential
\n" ); document.write( "function. Otherwise, you will make unnecessary calculations to find the coefficient \"k\"
\n" ); document.write( "of the exponential function, and, at the same time, you will demonstrate
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\n" ); document.write( "\n" ); document.write( "The second error in the solution by @josgarithmetic is that he uses given 60.4%
\n" ); document.write( "as the remaining mass, although the problem clearly states that it is the lost mass percentage.\r
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\n" ); document.write( "\n" ); document.write( "         Everything related to the solution procedure for exponential equation,
\n" ); document.write( "         when the half life is given, is described and clearly explained
\n" ); document.write( "         in my lesson/lessons, to which I referred in my post above.\r
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\n" ); document.write( "\n" ); document.write( "After my post, @josgarithmetic changed/corrected his calculations, related to the remaining mass.\r
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