Question 1189645
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When organisms die, the amount of carbon-14 in its system starts to decrease. 
The {{{highlight(half-life)}}} of the carbon-14 is about 6000 years. 
If a piece of human bone was found to contain only 1/3 of carbon-14 it originally had, how long ago did the human die ?
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<pre>
The remaining mass M(t) of Carbon-14 is exponentially decreasing function of the time.

According to the problem, the half-life is about 6000 years (5900 years, actually).


So we can write

    M(t) = {{{M(0)*(1/2)^(t/6000)}}}.


We want to find "t", when M(t) = {{{(1/3)*M(0)}}}. So we write the decay equation in the form

    {{{(1/3)*M(0)}}} = {{{M(0)*(1/2)^(t/6000)}}}.


Our goal is to find "t" from this equation. 


So, we divide both sides by {{{M(0)}}}, the initial mass of the Carbon-14, and we get

    {{{1/3}}} = {{{(1/2)^(t/6000)}}}.


Take logarithm base 10 from both sides

    {{{log((1/3))}}} = {{{(t/6000)*log((1/2))}}}

    t = {{{6000*(log((1/3))/log((1/2)))}}} = 9509.8 years, or 9500 years, rounded.    <U>ANSWER</U>
</pre>

Solved.


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On radioactive Carbon-14 dating, &nbsp;see and learn from the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/logarithm/Radioactive-decay-problems.lesson>Radioactive decay problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/logarithm/Carbon-dating-problems.lesson>Carbon dating problems</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Logarithms</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.