The half-life of a certain radioactive element is 800 years. How old is an object if only 12.5% of the radioactive atoms in it remain?
a
2400 years
b
1600 years
c
6400 years
To find the age of the object, we can use the formula for exponential decay:
N = N0 * (1/2)^(t/h)
Where:
N is the remaining amount
N0 is the initial amount
t is the time that has passed
h is the half-life of the radioactive element
Given that only 12.5% of the radioactive atoms remain, the remaining amount (N) is 0.125 times the initial amount (N0).
0.125 = 1 * (1/2)^(t/800)
To solve for t, we can take the logarithm of both sides:
log(0.125) = log[(1/2)^(t/800)]
Using the property of logarithms that log(a^b) = b * log(a), we can rewrite the equation as:
-3 = (t/800) * log(1/2)
Simplifying further:
-3 = (t/800) * (-0.3010)
-3 = -0.3010t/800
Multiplying both sides by 800:
-2400 = -0.3010t
Dividing both sides by -0.3010, we find:
t ≈ 2400 years
Therefore, the object is approximately 2400 years old.
a) 2400 years