Half-life of radioactive actinium

Question

The half-life of radioactive actinium (^227Ac) is 21.77 years. What percentage of a present amount of radioactive actinium will remain after 19 years?

Answer 1

Use 1 (= 100%) as the initial amount of the substance.

Then the new value = 1× (1/2)^(t/21.77)

= (1/2)^(19/21.77)

= 0.5461 (rounded to 4 decimal places). As a percent, this rounded value is 54.61%.

If the time was 21.77 years, the percent should be 50%, since one half-life would have elapsed. 19 years is almost 21.77, so we would expect something slightly more than 50% to be our answer. Clearly, our answer is in the right ballpark.