Final answer:
After 12 weeks, 31.25 mg of the original 500 mg sample of radioactive material would remain. This is calculated using the formula for radioactive decay after n half-lives, where n equals 4 in this case.
Step-by-step explanation:
The question asks us to determine the mass of a radioactive material that remains after a period of 12 weeks, given its half-life is 3 weeks. To do this, we calculate the number of half-lives that have occurred, and then apply a simple formula to find the remaining mass.
First, we determine the number of half-lives in 12 weeks:
Number of half-lives = Total time / Half-life duration = 12 weeks / 3 weeks per half-life = 4 half-lives
Next, we use the formula for radioactive decay after n half-lives:
Remaining mass = Initial mass × (½)^n
Where n is the number of half-lives.
Remaining mass of radioactive material = 500 mg × (½)^4
= 500 mg × (1/16)
= 31.25 mg
Therefore, after 12 weeks, 31.25 mg of the original 500 mg sample of radioactive material would remain, making Option 4 the correct answer.