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 $N=N0\ast e^{-\lambda t}$, 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 $\boxed{{\displaystyle 1523.1}}$ 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 $\boxed{{\displaystyle 1354.0}}$ days for a sample of 100 mg to decay to 54 mg.