If the half-life of a radioactive isotope is 32 days,how many half-lives must pass before a sample is less than ?

Question

If the half-life of a radioactive isotope is 32 days,how many half-lives must pass before a  sample is less than  ?

Quizplus · Accepted Answer

To solve this problem, we need to use the concept of exponential decay. The formula for calculating the amount of a radioactive substance remaining after a certain amount of time is:

N(t) = N0 * (1/2)^(t/T)

where:
N(t) = the amount of the radioactive substance remaining after time t
N0 = the initial amount of the radioactive substance
T = the half-life of the radioactive substance

We are given that the half-life of the radioactive isotope is 32 days, so T = 32. We want to find out how many half-lives must pass before the amount of the sample is less than half of its original amount. Mathematically, this can be expressed as:

N(t) < N0/2

Substituting the values given in the problem, we get:

N0 * (1/2)^(t/T) < N0/2

Simplifying this inequality, we get:

(1/2)^(t/T) < 1/2

Taking the logarithm of both sides (with base 1/2), we get:

t/T < 1

Solving for t, we get:

t < T

So, the amount of time it takes for the sample to decay to less than half of its original amount is less than one half-life. Therefore, the answer is C) 7, because after 7 half-lives, the amount of the sample remaining will be approximately 0.0078 times its original amount, which is less than 0.0125.

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