A wooden boat discovered just south of the Great Pyramid in Egypt has a carbon-14 to carbon-12 ratio that is 72.5% of that found in living organisms. How old is the boat?

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Understand that the problem involves radiocarbon dating, which uses the decay of carbon-14 to estimate the age of organic materials.

Recall that the half-life of carbon-14 is approximately 5730 years. This is the time it takes for half of the carbon-14 in a sample to decay.

Use the formula for radioactive decay: \( N(t) = N_0 \times (0.5)^{t/T} \), where \( N(t) \) is the remaining amount of carbon-14, \( N_0 \) is the initial amount, \( t \) is the time that has passed, and \( T \) is the half-life.

Set up the equation using the given ratio: \( 0.725 = (0.5)^{t/5730} \). This represents the fact that the carbon-14 to carbon-12 ratio is 72.5% of the original.

Solve for \( t \) by taking the natural logarithm of both sides and rearranging the equation to find the age of the boat.

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 ratio of carbon-14 to carbon-12 isotopes. Living organisms continuously exchange carbon with their environment, maintaining a constant ratio of these isotopes. Upon death, the carbon-14 begins to decay at a known rate (its half-life is about 5730 years), allowing scientists to estimate the time since the organism's death based on the remaining carbon-14.

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, this period is approximately 5730 years. Understanding half-life is crucial for calculating the age of an object based on the remaining amount of carbon-14, as it provides a consistent measure of decay over time.

Radioactive Decay and Ratios

Radioactive decay refers to the process by which unstable atomic nuclei lose energy by emitting radiation. The ratio of carbon-14 to carbon-12 in a sample can indicate how long it has been since the organism died. In this case, the boat's carbon-14 to carbon-12 ratio being 72.5% of that in living organisms suggests a specific amount of time has passed, which can be calculated using the known decay rate of carbon-14.

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Rate of Radioactive Decay

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A radioactive sample contains 1.55 g of an isotope with a halflife of 3.8 days. What mass of the isotope remains after 5.5 days?

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A sample of F-18 has an initial decay rate of 1.5⨉10⁵/s. How long will it take for the decay rate to fall to 2.5⨉10³/s? (F-18 has a half-life of 1.83 hours.)

Textbook Question

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.)