Final answer:
To find the fraction of a radioactive substance that remains after 113 years, given its half-life is 63 years, we calculate that approximately 1.79 half-lives have passed. The remaining fraction is calculated as (1/2)^1.79, which is approximately 0.292 or 29.2%.
Step-by-step explanation:
The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this question, we need to calculate what fraction of a radioactive substance with a half-life of 63 years will remain after 113 years.
We can use the formula for radioactive decay to find the number of half-lives that have passed. The formula is:
n = t / T1/2
where n is the number of half-lives, t is the total time elapsed, and T1/2 is the half-life of the substance. Using this, we find:
n = 113 years / 63 years = 1.79 half-lives (approximately)
The fraction of the substance remaining after n half-lives is given by:
(1/2)n = (1/2)1.79
Calculating the exponent we get approximately 0.292, which means around 29.2% of the original substance would remain after 113 years.