# Math Help - Could you help me a exponential decay question?

1. ## Could you help me a exponential decay question?

The question is "You buy a supposedly genuine 2000 year old eggshell of the extinct giant elephant bird Aepyornis, however you are not sure of its authenticity. You useyour Liquid Scintillation Counter to work out that the proportion of carbon-14, C14, in a sample of the eggshell is 0.883% of the proportion that would naturally occur at the time it was laid. Using methods from the notes, make a decision whether the eggshell authentic or not. The decay constant of C14 is r= 0.000125 per year."

I solved this by using M/M0 = e^rt, and r = 0.000125. So, e^(-0.000125*2000) = 0.778801....
C14 decreases as time increases. But, 0.883%, in other words, approximately 0.00883 of the eggshell was C14 as naturally occured.
If the eggshell is authentic, that means that the proportion of C14 has increased which yield contradiction. Thus, the eggshell is not authentic.

But, I wondered that 0.883% is supposed to be 88.3% or 0.883As the half-life of C14 is 5545 years, and the eggshell is around 2,000 years old, I guess 0.883% is too small for 2,000 years.

However, my lecturer replied No, there is no mistake. The percentage is supposed to be 0.883%.

Once you know this you should think about how to relate proportion to half-life and time elapsed.
I believe you're assuming that the original proportion between C^12 and C^14 might have been higher than it was, in which case you would be correct.
But there is no mistake.

So, I have no idea how I can solve it.

2. ## Re: Could you help me a exponential decay question?

I think you are correct. You would expect that after 2000 years the proportion of C14 remaining compared to its original concentartion would be 77.9%. So since the shell has only 0.83%, it must be much older than 2000 years. You can determine just how old it is using:

$\frac N {N_0} = e^{-\lambda t}$, so:

$t = - \frac {\ln ( \frac N {N_0} ) } {\lambda} = - \frac {\ln ( 0.00833 ) } {0.000125} = 38,303$ years.

The only other thing I can think of is that perhaps there is a typo in the problem statement. If the ratio of C14 was 0.833 instead of 0.833%, it would lead to an age of about 1462 years. Seems to me that's a more "reasonable" result than 38,300 years.

3. ## Re: Could you help me a exponential decay question?

I deeply appreciate it.
You saved me.
Thanks a lot !
Have a great day!