The half-life of radon-222 is 2.8 days. A homeowner used a radon test-kit to sample the air in his home, but forgot to send it for processing for 30 days. If the level of radon was actually 100 counts, what value would be reported by the test lab?

Question

BlackTom AI · Accepted Answer

First, I restate the problem to ensure clarity: the half-life of radon-222 is 2.8 days, the sample sat in the kit for 30 days before processing, and the actual radon level was 100 counts. We need the value reported after 30 days of decay.
Option 1: 9.3 counts. This would require far less decay than actually occurs; specifically it would imply about a 90% reduction, which is inconsistent with 30 days given the 2.8-day half-life.
Option 2: 0.060 counts. This aligns with the exponential decay model. The number of half-lives elapsed is 30 / 2.8 ≈ 10.714. The remaining fraction is (1/2)^{10.714}. Calculating: (1/2)^{10.714} = e^{ln(1/2) * 10.714} ≈ e^{-0.6931 * 10.714} ≈ e^{-7.425} ≈ 0.000595. Multiplying by the initial 100 counts gives about 0.0595 counts, which rounds to 0.060 counts. So this option matches the physics of exponential decay.
Option 3: 17 counts. This would imply far less decay than expected; it would correspond to a fraction around 0.17, which is inconsistent with the multi-half-life time elapsed (over 10 half-lives have passed), since multiple halvings drastically reduce the count well below 1% of the original.
Option 4: 1.5 counts. While it represents decay, it would require a smaller elapsed time or a different model; given 30 days and a 2.8-day half-life, the remaining counts should be far below 1.5, specifically around 0.06 counts as computed.
Option 5: 94 counts. This would indicate almost no decay, which contradicts having gone through more than 10 half-lives in 30 days; it’s physically implausible for this scenario.
In summary, the correct interpretation uses exponential decay with multiple half-lives elapsed, yielding about 0.060 counts after 30 days. The other options misrepresent the decay dynamics or miscalculate the remaining fraction after many half-lives.

The half-life of radon-222 is 2.8 days. A homeowner used a radon test-kit to sample the air in his home, but forgot to send it for processing for 30 days. If the level of radon was actually 100 counts, what value would be reported by the test lab? 单项选择题

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