20.113 The bromine isotope $\mathrm{Br}-75$ is used for imaging in positron emission tomography. Br-75 has a half-life of 57 h. How much time must elapse for a given dose of Br-75 to drop to $25 \%$ of its initial amount?

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∻Solution∻ ‖ 1 ⋮ **Determine the number of half-lives**: To find out how many half-lives are needed for the substance to drop to 25% of its initial amount, we use the relationship that after one half-life, the amount of substance is reduced to 50%, and after two half-lives, to 25%. ‖ ‖ 2 ⋮ **Calculate the total time**: Since the half-life of $\mathrm{Br}-75$ is 57 hours, and we have determined that two half-lives are needed to reach 25%, we multiply the number of half-lives by the half-life duration: $2 imes 57\ ext{hours}$. ‖ ∻Answer∻ ⚹ 114 hours ⚹ ∻Key Concept∻ ⚹ Half-life is the time required for a quantity to reduce to half its initial value. ⚹ ∻Explanation∻ ⚹ The time for a radioactive substance to drop to 25% of its initial amount is two half-lives, which is 114 hours for $\mathrm{Br}-75$. ⚹◊What is the mathematical formula used to calculate the amount remaining of a radioactive isotope after a certain amount of time has passed?⍭ Generate me a similar question◊

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