Final answer:
To calculate the remaining amount of polonium-210 after 415.2 days, we use the formula for exponential decay. After determining the number of half-lives that have passed, we find that 2.5 mg of polonium-210 will remain from the initial 20 mg given its half-life of 138.4 days.
Step-by-step explanation:
To calculate how many milligrams of polonium-210 remain after a certain period given the half-life, we use the formula for exponential decay based on half-lives. The formula to find the final amount (A) of a substance after a certain time (t) is given by: A = P(0.5)^(t/T), where P is the initial amount of the substance, t is the elapsed time, and T is the half-life of the substance.
Given:
- Initial amount, P = 20 mg
- Time elapsed, t = 415.2 days
- Half-life, T = 138.4 days
We can calculate the number of half-lives by dividing the time elapsed by the half-life of the substance: Number of half-lives = 415.2 days / 138.4 days = 3
With this, we can calculate the remaining amount of polonium-210:
Remaining amount, A = 20 mg * (0.5)^(3)
Therefore, after completing 3 half-lives:
A = 20 mg * 0.125 = 2.5 mg
So, 2.5 mg of polonium-210 will remain after 415.2 days.