Final answer:
After 80 years, approximately 12.5% of the radioactive material with a half-life of 32 years will remain.
Step-by-step explanation:
The half-life of a radioactive material is the amount of time it takes for 50% of its radioactivity to decay.
If a material has a half-life of 32 years, we can determine how much of it remains after 80 years by calculating the number of half-lives that have passed and then applying the decay process sequentially for each half-life.
After the first 32 years, half of the material remains (50% of the original).
After the second 32 years (64 years in total), this amount is halved again, resulting in 25% of the original amount.
After 80 years, which encompasses two full half-lives and an additional fraction of a half-life (80 - 64 = 16 years), the further reduction of the remaining material is accounted for.
We can summarize the process as follows:
First half-life (32 years): 50% remains.
Second half-life (64 years): 50% of 50% remains = 25%.
For the next 16 years: halfway between the 2nd and 3rd half-life, we need to halve the remaining percentage once more to find how much remains after 80 years.
So, 25% / 2 = 12.5% of the original material remains radioactive.
Therefore, after 80 years, approximately 12.5% of the radioactive material remains.