The half-life of Palladium-100 is 4 days, which means that every 4 days, the amount of Palladium-100 in the sample is reduced by half.
Let's start by finding how many half-lives have passed after 16 days.
16 days / 4 days per half-life = 4 half-lives
This means that the initial mass of the sample was doubled 4 times, since each half-life cuts the mass in half.
So, if the current mass is 2 mg, the initial mass would be:
Initial mass = 2 mg * 2^4 = 32 mg
Therefore, the initial mass of the sample was 32 mg.
To find the mass 7 weeks after the start, we need to find how many half-lives have passed in 7 weeks.
7 weeks = 7 * 7 days per week = 49 days
49 days / 4 days per half-life = 12.25 half-lives
This means that the amount of Palladium-100 in the sample would be reduced to:
Final mass = Initial mass * (1/2)^(12.25)
Final mass = 32 mg * 0.0566
Final mass ≈ 1.8112 mg
Therefore, the mass 7 weeks after the start would be approximately 1.8112 mg.