Question 1199250
.
The half-life of a radioactive substance is one hundred twenty-five days. How many days will it take for eighty-two percent of the substance to decay?
A. 180
B. 367
C. 275
D. 310
E. 532
F. 426
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<pre>
The half-life is given, so I write the exponential decay function 

in the base  {{{1/2}}}  in terms of half-lives  (in a way as it {{{highlight(highlight(SHOULD))}}} be done based on given info)


    m(t) = {{{m[0]*(1/2)^(t/125)}}},    (1)


where m(t) is the current remaining mass, {{{m[0]}}} is the starting mass and t is the time, in days.


We are given that m(t) = {{{(1-0.82)*m[0]}}} = {{{0.18*m[0]}}}, so  {{{m(t)/m[0]}}} = 0.18.


Therefore, equation (1) takes the form

    {{{(1/2)^(t/125)}}} = 0.18.


We want to find "t" from this equation.  For it, transform it step by step.
First step is to take logarithm base 10 of both sides

    {{{(t/125)*log((0.5))}}} = log(0.18)


From it, express t and calculate

    t = {{{125*(log((0.18))/log((0.5)))}}} = 309.24 days.


309.24 days is the required "continuous" time (real value).
If you want to get the answer as an integer number of days with a necessary margin, 
round it to the closest greater integer number, which is 310.
</pre>

Solved.


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<H3>Post-solution note :</H3> 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;when the half-life is given, write the exponential decay function in terms of half-life

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;with the base &nbsp;&nbsp;{{{1/2}}}.  &nbsp;&nbsp;It is the commonly accepted traditional &nbsp;&nbsp;GOOD &nbsp;STYLE.


Writing in other form &nbsp;(ekt-form), &nbsp;as Edwin makes it, &nbsp;&nbsp;IS &nbsp;NOT &nbsp;a good style when half-life is given.



Writing equation in the right form is not only the choice of the style.


Doing this way as I did, &nbsp;you perform only those calculations,
that are &nbsp;REALLY &nbsp;necessary, &nbsp;and do not make &nbsp;UNNECESSARY &nbsp;calculations.



It is the major reason of choosing that or the other style/approach.



///////////////



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.