The half life of a certain radioactive element is 1,250 years. what percent of the atoms remain after 7,500 years?
A. 25%
B. 1.5625%
C. 6.25%
D. 50%
The formula for calculating the percent of atoms remaining after a certain number of half-lives is:
Percent remaining = (1/2)^(number of half-lives) * 100%
In this case, since the half-life of the element is 1,250 years, we can calculate the number of half-lives that have passed in 7,500 years:
Number of half-lives = (total time) / (half-life) = 7500 / 1250 = 6
Now, we can substitute this value into the formula to find the percentage remaining:
Percent remaining = (1/2)^(6) * 100%
Percent remaining = (1/64) * 100% = 1.5625%
Therefore, the answer is B. 1.5625%.
To determine the percent of atoms that remain after 7,500 years, we need to calculate the number of half-lives that have occurred.
The formula to calculate the number of half-lives is:
Number of Half-Lives = Time Passed / Half-Life
In this case, the time passed is 7,500 years, and the half-life is 1,250 years.
Number of Half-Lives = 7,500 / 1,250 = 6
Now, we can calculate the percent of atoms that remain after 6 half-lives. The formula to calculate the remaining percent is:
Percent Remaining = (1/2)^(Number of Half-Lives) * 100
Using 6 as the number of half-lives, we have:
Percent Remaining = (1/2)^6 * 100
Calculating this expression:
Percent Remaining = (1/64) * 100
Percent Remaining = 1.5625%
Therefore, the correct answer is B. 1.5625%.
To determine the percent of atoms remaining after a certain amount of time, we can use the formula for exponential decay:
N(t) = N0 * (½)^(t / T)
Where:
- N(t) represents the number of atoms remaining at time t
- N0 represents the initial number of atoms
- '½' represents the fraction of atoms remaining after one half-life
- t represents the time elapsed
- T represents the half-life of the radioactive element
In this case, the half-life (T) is 1,250 years and the time elapsed (t) is 7,500 years.
To calculate the percent of atoms remaining, we need to find the ratio of N(t) to N0 and express it as a percentage.
N(t) / N0 = (½)^(t / T)
N(t) / N0 = (½)^(7,500 / 1,250)
= (½)^6
N(t) / N0 = 1/64
To express this ratio as a percentage, we multiply it by 100:
N(t) / N0 = (1/64) * 100
= 1.5625%
Therefore, the percent of atoms remaining after 7,500 years is approximately 1.5625%. The correct answer is B.