To determine how much Actinium-226 remains after 87 hours, we first need to calculate how many half-lives are contained within that time period.
The half-life of Actinium-226 is 29 hours. We can find the number of half-lives in 87 hours by dividing:
\[ \text{Number of half-lives} = \frac{87 \text{ hours}}{29 \text{ hours/half-life}} \approx 3 \]
Now, since we have approximately 3 half-lives, we can apply the formula for exponential decay:
\[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^{n} \]
where \( n \) is the number of half-lives.
Starting with 80 mg of Actinium-226:
\[ \text{Remaining amount} = 80 \text{ mg} \times \left(\frac{1}{2}\right)^3 \]
Calculating \( \left(\frac{1}{2}\right)^3 \):
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
Now calculate:
\[ \text{Remaining amount} = 80 \text{ mg} \times \frac{1}{8} = \frac{80}{8} = 10 \text{ mg} \]
Thus, after 87 hours, there will be 10 mg of Actinium-226 remaining.