Question

If a radioactive element has a half life of 4 hours, then x grams of the element dwindles to x/2 grams after 4 hours. If a nuclear reactor has 800 grams of that radioactive element, find the amount radioactive material after 16 hours.

Answer 1

The half-life of the radioactive element is 4 hours. This means that every 4 hours, the amount of the element will decrease by half.

Given:
- Initial amount of the element = 800 grams
- Half-life = 4 hours

To find the amount of radioactive material after 16 hours, we can calculate how many half-lives fit into 16 hours:

\[
\text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{16 \text{ hours}}{4 \text{ hours}} = 4
\]

Now, we can apply the half-life concept. After each half-life, the amount of material is multiplied by \(\frac{1}{2}\). Therefore, after 4 half-lives, the amount of radioactive material can be calculated using the formula:

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

Substituting the values:

\[
\text{Remaining amount} = 800 \text{ grams} \times \left(\frac{1}{2}\right)^4
\]

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

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

Now, calculate the remaining amount:

\[
\text{Remaining amount} = 800 \text{ grams} \times \frac{1}{16} = 50 \text{ grams}
\]

Thus, the amount of radioactive material after 16 hours is **50 grams**.