SOLUTION: The Mesopatamian civilization was dated by using the carbon-14 dating. A piece of wood discovered in an archaeological dig was found to have lost 62% of its carbon-14. Determine it

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Question 38296: The Mesopatamian civilization was dated by using the carbon-14 dating. A piece of wood discovered in an archaeological dig was found to have lost 62% of its carbon-14. Determine its age.
The answer is 7997 years but I don't know how to get to that answer.
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
you need to know the halflife of C-14. It is 5730 years.

Formula for half-life, T is +T+=+%28ln%282%29%29%2F%28y%29+ where y is the "time constant".

We need to figure out the time constant first:
+5730+=+%28ln%282%29%29%2Fy+
--> +y+=+%28ln%282%29%29%2F%285730%29+
y = 0.000120968 years^(-1)

Right then: exponential decay has a generic formula:
+N+=+N%5B0%5D%2Ae%5E%28-yt%29+
where N is the amount left "now" and +N%5B0%5D+ is the original amount.

+N%2FN%5B0%5D+=+e%5E%28-yt%29+

Now, we are told that the carbon14 has reduced by 62%. So, if there was originally 100g of it, there now is 38g... a reduction of 62%. So, we have:

+%2838%2F100%29+=+e%5E%28-yt%29+
+0.38+=+e%5E%28-yt%29+
+ln%280.38%29+=+-yt+
+t+=+%28ln%280.38%29%29%2F%28-y%29+

--> +t+=+%28ln%280.38%29%29%2F%28-0.000120968%29+

Working this out gives you 7998.7 years, which is in agreement with your answer, assuming rounding errors and the value of the half life used by the question.

jon.