Attempt 1
Yuestion 11 (') points)
A $200\mathrm{g}$ sample of radioactive polonium-210 has a half-life of 138 days. This means that every 138 days, the amount of polonium left in the sample is half of the original amount.
a) Write an equation that models the amount of Polonium
b) What mass of Polonium-210 is left after 2 years?
c) Determine how long it will take for the sample to reduce to $150\mathrm{g}$
11.

Step 1 ：The problem provides us with a $200\mathrm{g}$ sample of radioactive polonium-210 which has a half-life of 138 days. This means that every 138 days, the amount of polonium left in the sample is half of the original amount.

Step 2 ：We are asked to write an equation that models the amount of Polonium. The formula for exponential decay is $A=A0\ast (1/2{)}^{t/h}$, where $A$ is the final amount, $A0$ is the initial amount, $t$ is the time elapsed, and $h$ is the half-life.

Step 3 ：We are asked to find the mass of Polonium-210 left after 2 years. To do this, we need to convert the time from years to days. 2 years is equivalent to 730 days.

Step 4 ：Substituting the given values into the formula, we have $A0=200$, $t=730$, and $h=138$.

Step 5 ：Calculating the final amount $A$, we get $A=200\ast (1/2{)}^{730/138}=5.112401412165871$

Step 6 ：Rounding off to two decimal places, the mass of Polonium-210 left after 2 years is approximately $\boxed{{\displaystyle 5.11\text{ g}}}$.

Step 7 ：We are also asked to determine how long it will take for the sample to reduce to $150\mathrm{g}$. This requires solving the decay formula for $t$, with $A=150$.