Tritium (half-life = 12.3 y) is used to verify the age of expensive brandies. If an old brandy contains only the following fraction of the tritium present in new brandy, how long ago was it produced?
1/16
(1/2)^x=1/16
12.3x=?
i don't think it 4
x=4
12.3(4)=?
1/16 = (1/2)^4 so four half lives
4 * 12.3 as TutorCat said
To determine how long ago the old brandy was produced based on the fraction of tritium present, we need to use the concept of half-life.
The half-life of tritium is given as 12.3 years, which means that every 12.3 years, half of the tritium in a sample will decay.
In this case, the old brandy contains only 1/16th of the tritium present in new brandy.
Since we know that half of the tritium decays every 12.3 years, we can determine the number of half-lives that have occurred by finding the exponent that results in 1/16 when raised to that power.
1/2^x = 1/16
To solve for x, we can take the logarithm of both sides of the equation. Using the logarithm base 2 (because we have a power of 2 involved), we get:
log₂(1/2^x) = log₂(1/16)
Applying the logarithm property that shifted the exponent to the front, we have:
-x * log₂(1/2) = log₂(1/16)
The logarithm base 2 of 1/2 is -1, so the equation becomes:
-x * (-1) = log₂(1/16)
Simplifying further, we have:
x = log₂(16)
Using the logarithm base 2 identity, we know that 2^4 = 16. Therefore, log₂(16) = 4.
Thus, x = 4, which means that 4 half-lives have occurred.
Finally, we can determine the age of the old brandy by multiplying the number of half-lives by the half-life of tritium:
Age = 4 * 12.3 years = 49.2 years
Therefore, the old brandy was produced approximately 49.2 years ago.