A radioactive isotope has a half-life of 10 years. If you start with a 200-gram sample, how much of the substance will remain after 30 years?

Question

BlackTom AI · Accepted Answer

To solve, I start by identifying how many half-lives occur in 30 years given a 10-year half-life. Each half-life halves the amount present.
Option 12.5 g: This would require four half-lives (200 g -> 100 g -> 50 g -> 25 g -> 12.5 g). Since 30 years is only three half-lives, this amount corresponds to 30 years minus an extra half-life, so it’s not correct for 30 years.
Option 50 g: After two half-lives (200 g -> 100 g -> 50 g) you would reach 50 g, which corresponds to 20 years, not 30. Therefore this option describes a shorter time than asked.
Option 100 g: This is what remains after one half-life (200 g -> 100 g), i.e., after 10 years. It does not match the 30-year time point, so it’s incorrect.
Option 25 g: After three half-lives (30 years total), the amount is 200 g × (1/2)^3 = 200 g × 1/8 = 25 g. This aligns with the 30-year interval and the given half-life, making it consistent with the decay process.
In summary, each incorrect option corresponds to a different number of half-lives, and only the three-half-life scenario matches the 30-year time frame. The reasoning hinges on applying successive halvings for each 10-year period and checking which total time matches 30 years.

A radioactive isotope has a half-life of 10 years. If you start with a 200-gram sample, how much of the substance will remain after 30 years?

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