An ancient skull has a carbon-14 decay rate of 0.85 disintegrations per minute per gram of carbon (0.85 dis/min/g C). How old is the skull? (Assume that living organisms have a carbon-14 decay rate of 15.3 dis/min/g C and that carbon-14 has a half-life of 5715 years.)

Verified step by step guidance

1. Identify the initial and final decay rates: The initial decay rate for living organisms is 15.3 dis/min/g C, and the final decay rate for the skull is 0.85 dis/min/g C.
2. Use the decay formula: The decay of carbon-14 can be described by the formula \( N = N_0 e^{-\lambda t} \), where \( N \) is the final decay rate, \( N_0 \) is the initial decay rate, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.
3. Calculate the decay constant \( \lambda \): The decay constant is related to the half-life \( t_{1/2} \) by the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \). Substitute the given half-life of 5715 years to find \( \lambda \).
4. Rearrange the decay formula to solve for \( t \): \( t = \frac{\ln(N_0/N)}{\lambda} \). Substitute the values for \( N_0 \), \( N \), and \( \lambda \) into this equation.
5. Calculate \( t \): This will give you the age of the skull in years.