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# Calculus/Exponential Growth and decay.

Question
In 1947, earthenware jars containing what are know as Dead Sea Scrolls were found. Analysis showed that the scroll wrapping contained 76% of their original carbon-14, which has a half-life of 5750 years. What is the age of the Dead Sea Scrolls to the nearest year?

Questioner: Josey
Category:Calculus
Private:No
Subject:Math Calculus

Question:
In 1947, earthenware jars containing what are know as Dead Sea Scrolls were found. Analysis showed that the scroll wrapping contained 76% of their original carbon-14, which has a half-life of 5750 years. What is the age of the Dead Sea Scrolls to the nearest year?

A radioisotope will decay according to:

X = X0 exp(- k t), where

X is the amount we have now
X0 is the original amount
k is a constant that can be computed from the half-life, which we shall do.

After one 'half-life' of N years. X = X0/2 so:

X0/2 = X0 exp(- k N)

1/2 = exp(- k N)

2 = exp(k N)

k N = ln 2

k = ln 2 / N

N is the half-life, in this case 5760

Our scrolls have  0.76 X0 of C14 left

0.76 X0 = X0 exp( - (ln 2 / 5760) t)

0.76 = exp( - (ln 2 / 5760) t)

OK, you can take it from there, solving for t.

I get 2280.5 years

Calculus

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