Aluminum-30 has a half-life of 3.6 seconds.
a) What percentage of an initial sample will remain after 10 seconds?
b) What percentage of an initial sample will remain after 10 minutes?
a) The decay of a radioactive substance can be modeled by the equation N(t) = N₀(1/2)^(t/h), where N₀ is the initial amount, t is the time, and h is the half-life. In this case, N₀ = 100% and h = 3.6 seconds.
After 10 seconds, the equation becomes N(10) = 100%(1/2)^(10/3.6).
Calculating this value gives N(10) = 100%(0.5)^2.7778 ≈ 26.53%.
Therefore, approximately 26.53% of the initial sample will remain after 10 seconds.
b) To find the percentage of the initial sample that remains after 10 minutes, it is necessary to convert minutes into seconds. There are 60 seconds in a minute, so 10 minutes is equal to 10(60) = 600 seconds.
Using the same formula, N(600) = 100%(0.5)^(600/3.6).
Calculating this value gives N(600) = 100%(0.5)^166.67 ≈ 0.0294%.
Therefore, approximately 0.0294% of the initial sample will remain after 10 minutes.