A mammoth skeleton has a carbon-14 decay rate of 0.48 disintegration per minute per gram of carbon (0.48 dis/min • g C). When did the mammoth live? (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 yr.)

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Identify the given values: the current decay rate of the mammoth skeleton is 0.48 dis/min \( \cdot \) g C, the decay rate of living organisms is 15.3 dis/min \( \cdot \) g C, and the half-life of carbon-14 is 5715 years.

Use the formula for radioactive decay: \( N = N_0 \times e^{-\lambda t} \), where \( N \) is the current decay rate, \( N_0 \) is the initial decay rate, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

Calculate the decay constant \( \lambda \) using the half-life formula: \( \lambda = \frac{\ln(2)}{\text{half-life}} \).

Rearrange the decay formula to solve for \( t \): \( t = \frac{\ln(N_0/N)}{\lambda} \).

Substitute the known values into the rearranged formula to calculate the time \( t \) when the mammoth lived.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Carbon-14 Dating

Carbon-14 dating is a radiometric dating method used to determine the age of organic materials by measuring the amount of carbon-14 they contain. Living organisms absorb carbon-14 from the atmosphere, and when they die, the carbon-14 begins to decay at a known rate, characterized by its half-life. This technique is particularly useful for dating materials up to about 50,000 years old.

Half-Life

The half-life of a radioactive isotope is the time required for half of the isotope in a sample to decay into a different element or isotope. For carbon-14, the half-life is approximately 5,715 years. Understanding half-life is crucial for calculating the age of a sample based on the remaining amount of carbon-14 compared to its initial concentration in living organisms.

Disintegration Rate

The disintegration rate refers to the number of radioactive decays occurring per unit time, typically expressed in disintegrations per minute (dis/min). In this context, the disintegration rate of carbon-14 in living organisms is higher than in a deceased organism, allowing scientists to estimate the time since death by comparing the current decay rate to the known rate in living organisms.