Question 1191731
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The half life of cobalt -60 is 5.27 years. Starting with a sample of 150 mg, after how many years is 20 mg left?
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<pre>
In terms of the half-life, the general formula for radioactive decay of cobalt-60 is

    M(t) = {{{M(0)}}}.{{{(1/2)^(t/5.27)}}}

where M(t) is the current mass of the cobalt-60; M(0) is the initial mass,



Since 20 mg of the cobalt-60 remained, you have this equation

    20 = {{{150*(1/2)^(t/5.27)}}},  which reduces to   {{{20/150}}} = {{{(1/2)^(t/5.27)}}},

or

    0.133333 = {{{(1/2)^(t/5.27)}}}.
    


To solve it, take logarithm base 10 from both sides. You get an equation 

    {{{log(10,(0.133333))}}} = {{{(t/5.27)*log(10,(0.5))}}} .


Therefore,

     t = {{{5.27*(log(10,(0.133333))/log(10,(0.5)))}}} = 15.32 years  (rounded)


<U>ANSWER</U>.  It will happen in 15.32 years.
</pre>

Solved.



Regarding precision of the solution, &nbsp;if the half-life is given with two decimals, 

there is no sense to write or calculate the final sough time with the greater precision.


The greater precision is &nbsp;<U>a &nbsp;FICTION</U> &nbsp;in this case.


------------------


On radioactive decay, &nbsp;see the lesson

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

in this site.


You will find many similar &nbsp;(and different) &nbsp;solved problems there.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Use this lesson as your handbook, &nbsp;textbook, &nbsp;guide, &nbsp;tutorials, and &nbsp;(free of charge) &nbsp;home teacher.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Learn the subject from there once and for all.



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