Actinium-226 has a half-life of 29 hours. If 80 mg of Actinium-226 disintegrates over a period of 87 hours, how many mg of Actinium-226 will remain?(1 point)

Question

Responses

20 mg
20 mg

21.75 mg
21.75 mg

7.25 mg
7.25 mg

10 mg

Answer 1

To determine how much Actinium-226 remains after 87 hours, we can use the concept of half-lives.

1. Determine how many half-lives fit into the total time: The half-life of Actinium-226 is 29 hours. We need to find out how many half-lives are in 87 hours.

   \[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{87 \text{ hours}}{29 \text{ hours}} \approx 3 \]

   This means that 87 hours is about 3 half-lives (2.999, but we'll round it down for calculation purposes).

2. Calculate the remaining amount after 3 half-lives: The formula for the remaining amount after \( n \) half-lives is given by:

   \[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n \]

   Here, the initial amount is 80 mg, and \( n = 3 \):

   \[ \text{Remaining amount} = 80 \text{ mg} \times \left(\frac{1}{2}\right)^3 = 80 \text{ mg} \times \frac{1}{8} = 10 \text{ mg} \]

Thus, after 87 hours, 10 mg of Actinium-226 will remain.

The correct response is: 10 mg