The age of a document is in dispute, so archaeologists test for carbon-14. Due to radioactive decay, the amount A of carbon -14 present compared to the initial amount A0 after t years is given by the formula A(t) = A0e^-0.000124t . If 72% of the original amount of carbon- 14 is present in the document, how old is it?
.72 = 1 (e^-.000124t
-.000124t ln e = ln .72
but lne =1
t = ln .72 /-.000124
= appr 2649 years
check my arithmetic
To find the age of the document, we need to solve for t in the equation A(t) = A0e^(-0.000124t), where A(t) is the amount of carbon-14 present compared to the initial amount A0 after t years.
Given that 72% of the original amount is present, this means that A(t) = 0.72A0. We can substitute this value into the equation and solve for t.
0.72A0 = A0e^(-0.000124t)
Dividing both sides by A0:
0.72 = e^(-0.000124t)
To remove the exponential function, we can take the natural logarithm (ln) of both sides:
ln(0.72) = ln(e^(-0.000124t))
Using the property of logarithms, ln(e^x) = x:
ln(0.72) = -0.000124t
Now, we can solve for t by dividing both sides by -0.000124:
t = ln(0.72) / -0.000124
Using a scientific calculator, we can evaluate this expression:
t ≈ 5693.57 years
Therefore, the age of the document, based on the carbon-14 dating, is approximately 5693.57 years.