What formula is used for determining activity after a set amount of time?

The formula for determining activity after a set amount of time is based on the principles of exponential decay, which is characteristic of radioactive materials. The correct formula expresses how the remaining activity ( A(t) ) at time ( t ) is related to the initial activity ( A(0) ) and incorporates the half-life ( T(1/2) ) of the radioactive substance.

In this particular formula, ( A(t) = A(0) \exp\left(-\frac{\ln(2)}{T(1/2)} t\right) ), the term ( \exp(-\frac{\ln(2)}{T(1/2)} t) ) represents the decay of the radioactive substance over time. Here, ( \ln(2) ) is the natural logarithm of 2, which is used in the context of half-life, because after one half-life, the activity of the substance is reduced by half. The formula effectively models how the activity decreases exponentially as time progresses, allowing for accurate predictions of remaining activity based on elapsed time and the half-life.

The other options do not accurately express how to calculate the remaining activity over time based on the principles of radioactive decay. They