Question

Consider a quantity of a radioactive substance. The fraction of this quantity that remains after $t$ half-lives can be found by using the expression $3^{-t}$.

After how many half-lives will the fraction be $\frac{1}{243}$ of the original?

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Answer


We are given that the fraction of a radioactive substance that remains after $t$ half-lives can be determined using the expression $3^{-t}$.

Given: $$ \frac{1}{243} = 3^{-t} $$

We know that: $$ 243 = 3^5 $$ Thus, $$ \frac{1}{243} = \frac{1}{3^5} = 3^{-5} \quad \text{(since $\frac{1}{a^m} = a^{-m}$)} $$

By comparing both sides of the equation, we get: $$ t = 5 $$

So, after 5 half-lives, the fraction of the original quantity will be $\frac{1}{243}$.


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