A piece of wood discovered in an archaeological dig was found to have lost 62% of it's carbon-14. Carbon-14 has a half-life of 5630 years. Determine it's age.
I would use the equation
amount = initialvalue(1/2)^(t/half-life)
so
.62 = 1(1/2)^(t/5630)
ln .62 = (t/5630)ln .5
t/5630 = ln .62/ln .5 = .689659879
t = 3882.785
The age is appr. 3883 years
The tree's age can't be 3883 as the halflife is 5630 years and the tree has lost more than 50% of its c14, therefore, it would had to have undergone at least one halflife so the age would have to be at least more than the halflife length
To determine the age of the piece of wood, we can use the concept of half-life and the information provided.
The half-life of carbon-14 is given as 5630 years, which means that after this period, half of the carbon-14 present will decay.
We are given that the wood has lost 62% of its carbon-14. So, we need to figure out how many half-lives it took for this loss.
To do that, let's start with 100% of the carbon-14 in the wood. After one half-life (which is 5630 years), we would have 50% of the carbon-14 remaining. After the second half-life, we would have 25% remaining, and so on.
Since the wood has lost 62% of its carbon-14, we can write it as a fraction: 62/100 = 0.62.
Now, we need to find how many half-lives it took for the remaining carbon-14 to reach 0.62. We can use the formula:
Remaining fraction = (1/2)^(number of half-lives)
0.62 = (1/2)^(number of half-lives)
To solve for the number of half-lives, we can take the logarithm (base 1/2) of both sides:
log(0.62) = log[(1/2)^(number of half-lives)]
Using logarithmic properties, we can bring down the exponent:
log(0.62) = number of half-lives * log(1/2)
Now, we can calculate the number of half-lives:
number of half-lives = log(0.62) / log(1/2)
Using a calculator, the logarithm of 0.62 to the base 10 is approximately -0.206 and the logarithm of 1/2 to the base 10 is approximately -0.301.
number of half-lives = -0.206 / -0.301 = 0.685
Since we can't have a fraction of a half-life, we round the number of half-lives up to 1.
So, it took approximately 1 half-life (5630 years) for the wood to lose 62% of its carbon-14.
Therefore, the approximate age of the wood is 5630 years.