The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium- 100 has been reduced to a mass of $5 \mathrm{mg}$.

What was the initial mass (in $\mathrm{mg}$ ) of the sample?

What is the mass 8 weeks after the start?
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Step 1 ：Given that the half-life of Palladium-100 is 4 days, after 12 days a sample of Palladium-100 has been reduced to a mass of 5 mg.

Step 2 ：We can use the formula for exponential decay to find the initial mass of the sample. The formula is \(N_0 = N_t / (0.5 ^{t / T})\), where \(N_0\) is the initial mass, \(N_t\) is the final mass, \(t\) is the time elapsed, and \(T\) is the half-life.

Step 3 ：Substituting the given values into the formula, we get \(N_0 = 5 / (0.5 ^{12 / 4})\).

Step 4 ：Solving the equation, we find that the initial mass of the sample was \(N_0 = 40\) mg.

Step 5 ：Final Answer: The initial mass of the sample was \(\boxed{40}\) mg.