SOLUTION: The half-life of a radioactive material is about 34.3 years. How much of a 1-gram sample of the material is left after 20 years? (Round your answer to four decimal places.)

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Question 1162581: The half-life of a radioactive material is about 34.3 years. How much of a 1-gram sample of the material is left after 20 years? (Round your answer to four decimal places.)
Found 4 solutions by greenestamps, josgarithmetic, MathTherapy, ikleyn:
Answer by greenestamps(13216) About Me  (Show Source):
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The amount remaining, y, of an initial amount x, is
y+=+x%28%281%2F2%29%5En%29

where n is the number of half lives.

The number of half lives is the number of years, divided by the number of years in a half life.

We are given x=1g and we are to find y:

y+=+1%28%281%2F2%29%5E%2820%2F34.3%29%29

Use a calculator....

Very roughly, 20 years is something less than a half life, so your calculation should show an answer of slightly more than 0.5g remaining. If you get a much different answer, you aren't doing the calculation correctly.

Answer by josgarithmetic(39630) About Me  (Show Source):
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If using model, y=pe%5E%28-kx%29 for decay, x for time in years, p for amount at time 0, then for half life 34.3 years,

1%2F2=1%2Ae%5E%28-k%2A34.3%29
ln%281%2F2%29=-k%2A34.3
k=ln%282%29%2F34.3
k=0.0202


-

highlight_green%28y=1%2Ae%5E%28-0.0202x%29%29

for x=20 years, amount present then is e%5E%28-0.0202%2A20%29=highlight%280.68%29.

Answer by MathTherapy(10557) About Me  (Show Source):
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The half-life of a radioactive material is about 34.3 years. How much of a 1-gram sample of the material is left after 20 years? (Round your answer to four decimal places.)
Continuous GROWTH/DECAY formula: matrix%281%2C3%2C+A%2C+%22=%22%2C+A%5Bo%5De%5E%28kt%29%29

If half-life (.5-life) is 34.3 years, then DECAY CONSTANT, or matrix%281%2C3%2C+k%2C+%22=%22%2C+ln%28.5%29%2F34.3%29

We then get: matrix%281%2C3%2C+A%2C+%22=%22%2C+A%5Bo%5De%5Ekt%29

              ------- Substituting matrix%283%2C3%2C+1%2C+for%2C+A%5Bo%5D%2C+ln%28.5%29%2F34.3%2C+for%2C+k%2C+20%2C+for%2C+t%29

             Amount of the 1-gram material that remains after 20 years, or

Answer by ikleyn(52915) About Me  (Show Source):
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.

Since the input data is given in half-lives, the simplest way to solve the problem is to make all calculations base 2. 

The remaining mass formula is  M%5Bremaining%5D = M%5Bstart%5D%2A%281%2F2%29%5E%28t%2F34.3%29,  where t is the time in years.


Substituting t= 20 years, you have  20%2F34.3 = 0.5831 half-life times;  therefore


    M%5Bremaining%5D = 1%2F2%5E0.5831 = 0.6675  grams.

Solved.

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See the lesson
    - Radioactive decay problems
in this site.


The subject to learn from my post is THIS: 

    If input data is given in terms of half-life, the problem can be solved in these terms to the end with minimum efforts/calculations.


    It is NOT NECESSARY to convert data to ekt-model - on contrary, it is excessive and non-necessary work in this case.