The half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium100 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 4 weeks after the start?

Step 1 ：We are given that the half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium100 has been reduced to a mass of 5 mg. We are asked to find the initial mass of the sample.

Step 2 ：We know that the half-life of a substance is the time it takes for half of the substance to decay. This is an exponential decay problem. The formula for exponential decay is: \( N = N_0 * (1/2)^(t/h) \)

Step 3 ：In this formula: \(N\) is the final amount of the substance, \(N_0\) is the initial amount of the substance, \(t\) is the time that has passed, and \(h\) is the half-life of the substance.

Step 4 ：In this case, we know that \(N = 5mg\), \(t = 24 days\), and \(h = 4 days\). We want to find \(N_0\), the initial amount of the substance.

Step 5 ：We can rearrange the formula to solve for \(N_0\): \( N_0 = N / (1/2)^(t/h) \)

Step 6 ：Substituting the known values into the equation gives: \(N_0 = 5 / (1/2)^(24/4)\)

Step 7 ：Solving the equation gives: \(N_0 = 320.0\)

Step 8 ：Final Answer: The initial mass of the sample was \(\boxed{320 \, \text{mg}}\)