An unknown radioactive element decays into non-radioactive substances. In 880 days the radioactivity of a sample decreases by 33 percent.
(a) What is the half-life of the element?
half-life:
(days)
(b) How long will it take for a sample of $100 \mathrm{mg}$ to decay to $54 \mathrm{mg}$ ?
time needed:

Step 1 ：Given that the radioactivity of a sample decreases by 33 percent in 880 days, we can use this information to find the decay constant \(\lambda\). Once we have \(\lambda\), we can use it to find the half-life of the element using the formula: \(T = \frac{\ln(2)}{\lambda}\), where \(T\) is the half-life, and \(\ln\) is the natural logarithm.

Step 2 ：Using the decay formula \(N = N0 * e^{-\lambda t}\), where \(N\) is the final amount of the element, \(N0\) is the initial amount of the element, \(\lambda\) is the decay constant, and \(t\) is the time, we can calculate the decay constant \(\lambda = 0.0004550881438603696\).

Step 3 ：Substituting \(\lambda\) into the half-life formula, we get \(T = \frac{\ln(2)}{0.0004550881438603696}\), which simplifies to \(T = 1523.1053366501612\). Therefore, the half-life of the element is approximately \(\boxed{1523.1}\) days.

Step 4 ：Now, let's calculate how long it will take for a sample of 100 mg to decay to 54 mg. We can use the decay formula again, but this time we need to solve for \(t\): \(t = -\frac{\ln(N/N0)}{\lambda}\), where \(N0\) is the initial amount of the element (100 mg), \(N\) is the final amount of the element (54 mg), and \(\lambda\) is the decay constant (which we calculated earlier).

Step 5 ：Substituting the given values into the formula, we get \(t = -\frac{\ln(54/100)}{0.0004550881438603696}\), which simplifies to \(t = 1353.9929522155944\). Therefore, it will take approximately \(\boxed{1354.0}\) days for a sample of 100 mg to decay to 54 mg.