The original amount of a radioactive sample should be multiplied by which expression to calculate the amount of the sample that remains after $n$ half-lives have passed?
A. $(1 / 2) \times n$
B. $(1 / n)^2$
C. $(1 / 2)^n$
D. $1 /(2 n)$
Answer (2)
After each half-life, the amount of the radioactive sample is halved.
After n half-lives, the remaining amount is ( 2 1 ) n of the original amount.
Therefore, the original amount should be multiplied by ( 2 1 ) n .
The expression is ( 2 1 ) n .
Explanation
Understanding the Problem Let's analyze the problem. We are given a radioactive sample and we want to find the expression that tells us what fraction of the original sample remains after n half-lives. A half-life means that after each half-life, the amount of the sample is reduced by half.
Finding the Pattern Let's consider what happens after each half-life:
After 1 half-life, the remaining amount is 2 1 of the original amount.
After 2 half-lives, the remaining amount is 2 1 × 2 1 = ( 2 1 ) 2 of the original amount.
After 3 half-lives, the remaining amount is 2 1 × 2 1 × 2 1 = ( 2 1 ) 3 of the original amount.
We can see a pattern here. After n half-lives, the remaining amount is ( 2 1 ) n of the original amount.
Final Answer Therefore, the original amount of the radioactive sample should be multiplied by ( 2 1 ) n to calculate the amount of the sample that remains after n half-lives have passed.
Examples
Radioactive decay is used in carbon dating to determine the age of ancient artifacts. If we know the half-life of carbon-14 (approximately 5,730 years), we can use the formula ( 2 1 ) n to estimate how much carbon-14 should be remaining in a sample after a certain number of years, and thus estimate the age of the artifact. For example, if an artifact has gone through 2 half-lives of carbon-14, then the remaining carbon-14 is ( 2 1 ) 2 = 4 1 of the original amount.
The expression used to calculate the remaining amount of a radioactive sample after n half-lives is ( 2 1 ) n . This means the original amount should be multiplied by this expression to find how much remains. Therefore, the correct choice is C. ( 2 1 ) n .
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