Question 1190364
.
Carbon-14 is used to determine the age of artifacts. It has a half-life of 5700 years. This means that
in 0 years 100%, of the original amount of carbon-14 remains in the artifact. In 5700 years 50% of
the original amount of carbon-14 remains.
a) Suppose that 40% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.
b) Suppose that 75% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.
c) Suppose that 2% of the original carbon remains in a piece of pottery.
Estimate the age of the pottery.
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<pre>
You are given that half-life of C-14 is 5700 years.

So, you can write the decay function in the form

    m(t) = {{{m(0)*(1/2)^(t/5700)}}},    (1)


where m(0) is the original mass, m(t) is the current mass and t is the time, in years.


To answer all questions that follow, divide both sides of the decay equation (1) by m(0). You will get

    {{{m(t)/m(0)}}} = {{{(1/2)^(t/5700)}}}.        (2)


Now let's answer the questions, one after another.



(a)  You are given {{{m(t)/m(0)}}} = 40% = 0.4.

     So, equation (2) gives you

         {{{(1/2)^(t/5700)}}} = 0.4.


     Take logarithm base 10 of both sides

         {{{(t/5700)*log((0.5))}}} = log(0.4)

         t = {{{(5700*log((0.4)))/log((0.5))}}} = 7535  years.    <U>ANSWER</U>



(b)  You are given {{{m(t)/m(0)}}} = 75% = 0.75.

     So, equation (2) gives you

         {{{(1/2)^(t/5700)}}} = 0.75.


     Take logarithm base 10 of both sides

         {{{(t/5700)*log((0.5))}}} = log(0.75)

         t = {{{(5700*log((0.75)))/log((0.5))}}} = 2366  years.    <U>ANSWER</U>




(c)  You are given {{{m(t)/m(0)}}} = 2% = 0.02.

     So, equation (2) gives you

         {{{(1/2)^(t/5700)}}} = 0.02.


     Take logarithm base 10 of both sides

         {{{(t/5700)*log((0.5))}}} = log(0.02)

         t = {{{(5700*log((0.02)))/log((0.5))}}} = 32170  years.    <U>ANSWER</U>
</pre>

Solved.


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To solve the problem, you do not need make tons of unnecessary calculations, that @Theo makes in his post.


When you know the half-life, the way which I show you is the standard, shortest and straightforward way to solve the problem.



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There is a group of lessons in this site, which covers many similar problems on Carbon dating

&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;and

&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> 


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.