Question

You have 8,640 grams of a radioactive kind of cesium. If its half-life is 30 years, how much will be left after 120 years?

Answer 1

To determine how much of the cesium will be left after 120 years, we can use the concept of half-lives. The half-life of this radioactive cesium is 30 years, meaning that every 30 years, half of the substance will have decayed.

First, we calculate the number of half-lives that occur in 120 years:

\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{120 \text{ years}}{30 \text{ years}} = 4
\]

Next, we apply the half-life formula to find out how much cesium remains after 4 half-lives. The amount remaining can be calculated using the formula:

\[
\text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{\text{number of half-lives}}
\]

Substituting in the values:

\[
\text{Remaining amount} = 8640 \text{ grams} \times \left(\frac{1}{2}\right)^{4}
\]

Calculating \(\left(\frac{1}{2}\right)^{4}\):

\[
\left(\frac{1}{2}\right)^{4} = \frac{1}{16}
\]

Now we can plug that into the remaining amount calculation:

\[
\text{Remaining amount} = 8640 \text{ grams} \times \frac{1}{16} = 540 \text{ grams}
\]

After 120 years, there will be **540 grams** of the radioactive cesium left.