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. 3200 years
B. 1600 years
C. 6400 years
We can solve this problem by using the half-life formula:
N(t) = N₀ * (1/2)^(t/T)
where:
N(t) is the remaining percentage of radioactive atoms after time t
N₀ is the initial percentage of radioactive atoms (100% in this case)
t is the time that has passed
T is the half-life of the radioactive element
In this problem, we know that N(t) is 12.5% and T is 800 years. We want to find t.
12.5% = 100% * (1/2)^(t/800)
Divide both sides of the equation by 100%:
0.125 = (1/2)^(t/800)
Take the logarithm (base 2) of both sides to isolate t/800:
log₂(0.125) = t/800
Simplify the left side of the equation:
log₂(1/8) = t/800
Rewrite 1/8 as 2^(-3):
-3 = t/800
Multiply both sides of the equation by 800:
-3 * 800 = t
t = -2400
Since time cannot be negative, we discard the negative solution, so t = 2400 years.
Therefore, the object is 2400 years old.
The correct answer is not listed in the options provided.
To determine the age of the object, we can use the formula for exponential decay:
Nt = N0 * (1/2)^(t / T)
Where:
Nt is the final amount (in this case, 12.5% of the original amount),
N0 is the initial amount,
t is the time that has passed,
T is the half-life of the element.
Given that the half-life is 800 years and only 12.5% of the radioactive atoms remain, we can substitute these values into the formula:
Nt = N0 * (1/2)^(t / 800)
0.125 = 1 * (1/2)^(t / 800)
Now, we need to solve for t. To do this, we take the logarithm (base 2) of both sides:
log2(0.125) = log2(1 * (1/2)^(t / 800))
-3 = t / 800 * log2(1/2)
-3 = t / 800 * (-1)
t / 800 = 3
t = 3 * 800 = 2400 years
Therefore, the object is 2400 years old.
To find the age of the object, we can use the concept of half-life. The half-life of the radioactive element is given as 800 years, which means that after every 800 years, the amount of the radioactive element will reduce by half.
Given that only 12.5% of the radioactive atoms remain in the object, we can infer that the object has gone through multiple half-lives.
To calculate the number of half-lives, we need to find the fraction of radioactive atoms that remain. Since 12.5% can be written as 0.125 in decimal form, the fraction of radioactive atoms that remain is 0.125.
To calculate the number of half-lives, we use the formula:
Number of half-lives = (log(initial amount remaining) - log(final amount remaining)) / (log(2))
Since the initial amount remaining is 1 (100%) and the final amount remaining is 0.125 (12.5%), we can plug these values into the formula:
Number of half-lives = (log(1) - log(0.125)) / (log(2))
Number of half-lives = (0 - (-0.903)) / (0.301)
Number of half-lives = 0.903 / 0.301
Number of half-lives ≈ 3
This tells us that the object has gone through three half-lives.
To find the age of the object, we multiply the number of half-lives by the length of one half-life:
Age of the object = Number of half-lives * Half-life
Age of the object = 3 * 800 years
Age of the object = 2400 years
Therefore, the object is approximately 2400 years old.
None of the given options match the calculated age, so none of the provided answers are correct.