SOLUTION: A certain radioactive isotope has a​ half-life of approximately 78 years. How many years would be required for a given amount of this isotope to decay to 60​% of that​ amount

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Question 1162420: A certain radioactive isotope has a​ half-life of approximately 78 years. How many years would be required for a given amount of this isotope to decay to 60​% of that​ amount? ​(​Hint: first find k. Then use k to find​ t)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
MODEL Form: y=pe%5E%28-kx%29


If half-life 78 years, then k=ln%282%29%2F78.
.


When any given amount to go to 60% of the starting amount?
ln%280.6%29=-%28ln%282%29%2F78%29x------------solve this for x.

Answer by ikleyn(52792) About Me  (Show Source):
You can put this solution on YOUR website!
.

We are given that the half-life period is 78 years;  therefore, we can write


    p(t) = %281%2F2%29%5E%28t%2F78%29,      (1)


where p(t) is the remaining mass fraction. (It is the standard radioactive decay model in terms of half-life period).


The problem asks to determine the time "t" when p(t) = 0.6.


In this case, the equation (1) takes the form


    0.6 = %281%2F2%29%5E%28t%2F78%29.


Take logarithm base 2 of both sides


    log%282%2C%280.6%29%29%29 = %28t%2F78%29%2Alog%282%2C+%281%2F2%29%29


    t = %2878%2Alog%282%2C%280.6%29%29%29%2F%28%28-1%29%29 = %28-78%29%2Alog%282%2C%280.6%29%29 = 57.48 years.    ANSWER

Solved.

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    - Radioactive decay problems
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The lesson to learn from my post is THIS: 

    If you are given input data in terms of half-life, you do not need to convert your data 

    into ekt-model.  Such conversion is an excessive work and unnecessary calculations.


    You can complete all calculations in terms of the half-life model, working consistently with degrees of 2, 
    which is your base in this case..