radioactive Carbon 14 in a dead organism decays according to the equation A=A e^.000124t where t is in years and A0 is the amount at t=0. Estimate the age of a skull uncovered in an archeological site if 10% of the original Carbon 14 is still present.
A = Ao*e(^0.000124t)
If the final amount, A, is equal to 10% of initial amount, Ao, thus
0.1*Ao = Ao*e(^0.000124t)
0.1 = e(^0.000124t)
Get the ln of both sides:
ln (0.1) = ln(e(^0.000124t))
ln (0.1) = 0.000124t
Now solve for t. Just one clarification, I think the exponent of e should be negative if it's a decay problem. Because if not, the time that would be solved would be negative, and time cannot be negative.
Hope this helps~ `u`
To estimate the age of the skull, we need to use the given equation for radioactive decay:
A = A₀ * e^(0.000124t),
where A is the current amount of Carbon 14, A₀ is the original amount of Carbon 14, t is the time in years, and e is the mathematical constant approximately equal to 2.71828.
In this case, we are given that only 10% of the original Carbon 14 is still present. Therefore, we can set up the equation:
0.10A₀ = A₀ * e^(0.000124t).
Simplifying the equation by canceling out A₀ on both sides gives:
0.10 = e^(0.000124t).
To solve for t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(0.10) = ln(e^(0.000124t)).
Using the logarithmic property ln(e^x) = x, we get:
ln(0.10) = 0.000124t.
Now, we can isolate t by dividing both sides of the equation by 0.000124:
t = ln(0.10) / 0.000124.
Using a calculator to compute the right side of the equation, we find:
t ≈ 6,011.88 years.
Therefore, the estimated age of the skull is approximately 6,012 years.