Final answer:
It would take 458 years for half of the initial 1,000 atoms of americium-242 (assuming the intended isotope is americium-241, which is used in smoke detectors) to decay, as this is the half-life of americium-241.
Step-by-step explanation:
The subject of this question is Physics, specifically involving the concept of radioactive decay and half-life. The student is inquiring about the time it would take for 500 atoms of americium-242 to decay, given an initial quantity of 1,000 atoms.
To determine how long it takes for 500 of the 1,000 atoms of americium-242 to decay, which is half of the initial quantity, we use the principle that the time required for half the atoms of a radioactive isotope to decay is its half-life. The half-life of americium-241, which is commonly used in smoke detectors, is 458 years. Although the question refers to americium-242, for educational purposes, let's consider it as a typo since smoke detectors actually use americium-241. Assuming the question meant americium-241, it would take 458 years for 500 of the 1,000 atoms to decay, as this is its half-life.
For the reference information provided, the activity of the americium-241 in the smoke detectors can be calculated by considering the activity of 1 gram and scaling it down to the actual amount used in smoke detectors, which is 0.25 mg.