Final answer:
To find the percentage of actinium remaining after 19 years, apply the half-life formula and calculate using the provided half-life and the elapsed time. This will give you the remaining percentage that is not as intuitive as halving the amount due to the time being less than one full half-life.
Step-by-step explanation:
The half-life of a radioactive isotope, like actinium (²⁷⁷Ac), is the time it takes for half of the substance to decay. Actinium has a half-life of about 22 years, but we need to calculate the remaining amount after 19 years. To do this, we can use the formula: final amount = initial amount * (1/2)^(time elapsed / half-life). Since 19 years is less than one half-life, we cannot simply halve the quantity, and instead must use the formula to find the exact amount.
If the initial amount is 100%, and we substitute 19 years as the time elapsed and 22 years as the half-life, we get:
Remaining amount = 100% * (1/2)^(19/22)
Calculating this gives us the percentage of actinium that remains after 19 years.