The half-life of a certain radioactive element is 1,250 years. What percent of the atoms remain 7,500 years?
a
25%
b
1.5625%
c
50%
d
6.25%
The percent of atoms that remain after a given time can be calculated using the formula:
Percent Remaining = (1/2)^(t/h) * 100
Where:
t = time passed in years
h = half-life of the element in years
In this case, t = 7,500 years and h = 1,250 years. Plugging these values into the formula:
Percent Remaining = (1/2)^(7,500/1,250) * 100
Percent Remaining = (1/2)^6 * 100
Percent Remaining = (1/64) * 100
Percent Remaining = 1.5625%
Therefore, the correct answer is option b) 1.5625%.
To determine the percent of atoms that remain after a certain time, we need to calculate the number of half-lives that have passed.
Given that the half-life of the radioactive element is 1,250 years, we can calculate the number of half-lives that have passed by dividing the total time (7,500 years) by the half-life:
Number of half-lives = 7,500 years / 1,250 years = 6 half-lives
Since each half-life means that 50% of the atoms decay, we can calculate the percent of atoms that remain using the formula:
Percent remaining = (1/2)^(number of half-lives) * 100%
Plugging in the calculated number of half-lives:
Percent remaining = (1/2)^6 * 100%
Evaluating this expression:
Percent remaining = (1/64) * 100% = 1.5625%
Therefore, the correct answer is (b) 1.5625%.
To find the percent of atoms that remain after a certain period of time, we can use the concept of half-life. The half-life of a radioactive element is the time it takes for half of the atoms to decay. In this case, the half-life is 1,250 years.
To calculate the percent of atoms that remain after 7,500 years, we need to determine how many half-lives have passed. We can do this by dividing the given time (7,500 years) by the half-life (1,250 years).
Number of Half-lives = Time / Half-life = 7,500 years / 1,250 years = 6 half-lives
Each half-life reduces the amount of remaining atoms by half. After 6 half-lives, the fraction of atoms remaining would be (1/2)^6.
(1/2)^6 = 1/64 ≈ 0.015625
To express this as a percentage, we can multiply the fraction by 100.
0.015625 * 100 = 1.5625%
Therefore, the answer is option b: 1.5625% of the atoms remain after 7,500 years.