Problem 64 - Calculus for Biology and Medicine

Question

The half-life of \(\mathrm{C}^{14}\) is 5730 years. Suppose that wood found at an archeological excavation site contains about \(35 \%\) as much \(\mathrm{C}^{14}\) (in relation to \(\mathrm{C}^{12}\) ) as does living plant material. Determine when the wood was cut.

Step-by-Step Solution

Answer

The wood was cut approximately 8732 years ago.

Step 1: Understand the Concept of Half-life

The half-life of a substance is the time it takes for half of it to decay. For Carbon-14 (\(^\{14\}C\)), this period is 5730 years. This means that in 5730 years, only half of the original amount of \(^\{14\}C\) will remain.

Step 2: Use the Exponential Decay Formula

To find when the wood was cut, we use the exponential decay formula: \(N(t) = N_0 \times (\frac{1}{2})^{t/T}\), where \(N(t)\) is the current quantity of \(^\{14\}C\), \(N_0\) is the initial quantity, \(t\) is the time, and \(T\) is the half-life (5730 years for \(^\{14\}C\)).

Key Concepts

Exponential Decay Formula

The exponential decay formula is a useful mathematical tool to model situations where quantities decrease over time. This is particularly relevant for substances like Carbon-14, which is a radioactive isotope found in many organic materials. The general formula is:
\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T} \]
Here’s what each component means:

• \(N(t)\) is the quantity remaining after a certain time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(t\) represents the amount of time that has elapsed.
• \(T\) is the half-life of the substance, which is the time taken for half of the material to decay.

This formula allows us to calculate how much of a substance remains at any given time, given its half-life.
In our exercise, it aids in finding out when the tree was cut by comparing the remaining amount of Carbon-14 to its initial quantity.

Carbon-14 Dating

Carbon-14 dating, also known as radiocarbon dating, is a method for determining the age of an object containing organic material. It relies on the principle that Carbon-14, a radioactive isotope of carbon, is consistently being formed in the atmosphere. This isotope gets incorporated into all living organisms.
As living things die, they stop absorbing Carbon-14, and thus the existing Carbon-14 begins to decay. By measuring the remaining amount of Carbon-14 and using its known half-life (5730 years), scientists can estimate the time since the organism's death.
In the exercise, we know that the wood contains about 35% of the original Carbon-14 compared to a live plant. This percentage plays a role alongside the exponential decay formula to compute the time since the tree, from which the wood was cut, died.

Archeological Dating

Archeological dating is crucial for understanding the timeline in which historical artifacts and sites existed. Among various dating methods, Carbon-14 dating stands out for its ability to date organic materials reliably up to about 50,000 years.
In archeological excavations, finding out the age of artifacts like wooden tools or structures is often vital. Carbon-14 dating helps archeologists ascertain this by allowing them to calculate the time since these items were made.
In the context of the exercise, archeological dating through Carbon-14 measurement of the wood helps us conclude how long ago the tree was cut down. This gives insight into the historical period in which people might have interacted with the object, helping piece together timelines of human activity and environmental changes.