Estimate the age of the skull if the half-life of carbon-14 is about 5600 years.

Question

Archaeologists believe the age of the skull to be the same age as the campfire. It is determined that only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire.

Estimate the age of the skull if the half-life of carbon-14 is about 5600 years.

I have already tried the answer of 5600 * ln(0.02) / ln(0.5). I am not sure why that is incorrect if someone knows where I am going wrong that would be very helpful.

Answer 1

Hello. This is a two-stage process using P = P₀ e^-kt, where P is the final amount, P₀ is the initial amount, k is the constant of proportionality, and t is out the independent variable of time.

Stage 1:

After 5600 years, we know P = 1/2 * P₀. This allows us to solve for k.

1/2 * P₀ = P₀ e^{-k 5600} (divide both sides by P₀)

1/2 = e^{-k 5600} (take the natural log of both sides)

ln(1/2) = -k 5600 (divide both sides by -5600)

-ln(1/2) / 5600 = k (note that ln(1/2) = ln(1) - ln(2) = 0 - ln(2) = -ln(2))

-(-ln(2)) / 5600 = k

ln(2) / 5600 = k

Thus, f(t) = P₀ e^{(-ln(2) / 5600) * t} (recall there is a negative multiplying the k)

Stage 2:

Now we use the fact that P = 0.02 P₀ to solve for t.

0.02 P₀ = P₀ e^{(-ln(2) / 5600) * t} (divide both sides by P₀)

0.02 = e^{(-ln(2) / 5600) * t} (use the log rule of c*log_bx = log_bx^c, move the negative to the exponent of 2)

0.02 = e^{(ln(2^(-1)) / 5600) * t} (take the natural log of both sides)

ln(0.02) = (2^-1 / 5600 ) * t

ln(0.02) = t / (2¹ * 5600) (multiply both sides by 2*5600 = 11200)

11200 * ln(0.02) = t

Conclusion:

t is about 7763 years. We estimate the campfire and the skull to be about this age.