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The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with 165 grams of a radioactive isotope, how much will be left after 3 half-lives? Use the calculator provided and round your answer to the nearest gram. \( \square \) grams

Ask by West Herrera. in the United States
Jan 27,2025

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Answer

After 3 half-lives, 21 grams of the isotope will remain.

Solution

To determine how much of the radioactive isotope remains after 3 half-lives, follow these steps: 1. **Initial Amount:** Start with **165 grams** of the isotope. 2. **Half-Life Reduction:** Each half-life reduces the remaining amount by half. - **After 1 half-life:** \( \frac{165}{2} = 82.5 \) grams - **After 2 half-lives:** \( \frac{82.5}{2} = 41.25 \) grams - **After 3 half-lives:** \( \frac{41.25}{2} = 20.625 \) grams 3. **Rounding:** Round **20.625 grams** to the nearest whole number: **21 grams** **Final Answer:** \( \boxed{21} \) grams

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After 1 half-life, 165 grams will reduce to 82.5 grams. After the second half-life, that amount becomes 41.25 grams. Finally, after the third half-life, you will be left with about 20.625 grams. When rounding to the nearest gram, the final answer is 21 grams. Speaking of half-lives, did you know that the concept of half-life isn't just relegated to radioactive isotopes? It's also applied in medicine, particularly in pharmacology, to determine how long a drug stays active in the body. This information helps ensure that medications are used effectively and safely! On a fun note, the gravitational concept of a “half-life” can be seen in everyday phenomena! For example, consider how your leftovers lose their freshness over time. Just like isotopes decay, the taste and quality of food can diminish, letting you know when it's time to toss that takeout!

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