Problem 38 - Precalculus with Limits

Question

Radioactive Decay Carbon 14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \({ }^{14} \mathrm{C}\) absorbed by a tree growing today. A piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal, assuming that the half-life of \({ }^{14} \mathrm{C}\) is 5715 years?

Step-by-Step Solution

Answer

To see how long ago the tree was burned to create the charcoal, find the value of \(t\) in the last equation. This should give you the answer in years

Step 1: Identifying the given information

We have \(15\%\) of the \({ }^{14} \mathrm{C}\) remaining, meaning that \(15\% = 0.15\) is the remaining fraction of the initial amount. The half-life of \({ }^{14} \mathrm{C}\) is given as \(5715\) years.

Step 2: Setting up the decay model

We use the decay formula \(N(t) = N0 * (1/2)^{t/t_{1/2}}\) where \(N(t)\) is the remaining amount after time \(t\), \(N0\) is the initial amount, \(t_{1/2}\) is the half-life of the radioactive isotope, and \(t\) is the time of decay. We are not given the initial amount of \({ }^{14} \mathrm{C}\), but we know the amount remaining relative to the initial amount, which is 0.15. Thus, we can write the formula as \(0.15 = (1/2)^{t/5715}\)

Step 3: Solving for time

Firstly, take the natural logarithm (ln) on both sides to simplify the exponential equation. That gives us \(\ln(0.15)= t/5715 * \ln(1/2)\). Dividing both sides of the equation by \(\ln(1/2)\) results in \(\frac{\ln(0.15)}{\ln(1/2)} = \frac{t}{5715}\). By multiplying both sides by 5715 we get the amount of time that has passed as: \(t = 5715 * \frac{\ln(0.15)}{\ln(1/2)}\)

Step 4: Calculating the time that has passed

Doing the above calculation arrives at the answer. Remember that your answer should be in years because the half-life was given in years

Key Concepts

Carbon 14 Dating

Carbon 14 dating is a method used by scientists to determine the age of biological materials such as bones, wood, or charcoal. It relies on the principle that living organisms absorb carbon, including the radioactive isotope Carbon-14 (\r{\({ }^{14}C\)}). Upon death, the organism stops absorbing carbon, and the \r{\({ }^{14}C\)} begins to decay at a known rate.

By measuring how much \r{\({ }^{14}C\)} remains in the material compared to a modern reference, scientists can calculate when the organism died. This technique is extremely useful in archaeology and paleontology for dating artifacts and fossils up to about 50,000 years old. The accuracy of Carbon 14 dating depends on the assumption that the concentration of \r{\({ }^{14}C\)} in the atmosphere has remained constant over time, which is a basis generally held by the scientific community.

Decay Formula

The decay formula plays a crucial role in Carbon 14 dating and is fundamental in understanding radioactive decay as a whole. It represents how the quantity of a radioactive isotope diminishes over time.

The formula is expressed as \(N(t) = N0 * (1/2)^{t/t_{1/2}}\), where \(N(t)\) is the number of radioactive atoms remaining after time \(t\), \(N0\) is the initial number of radioactive atoms, \(t_{1/2}\) is the half-life of the isotope, and \(t\) is the elapsed time. By inserting the known values into this equation, you can solve for the unknown time variable. When faced with a decay problem, it's often helpful not to think in terms of specific amounts but rather in fractions or percentages, which simplifies the process since the initial quantity often cancels out.

Half-Life

Half-life is a term that describes the amount of time it takes for half of the radioactive atoms in a sample to undergo decay. This value is a constant for a given isotope and is not influenced by environmental conditions.

Understanding half-life is essential for interpreting decay formulas and Carbon 14 dating results. For \r{\({ }^{14}C\)}, the half-life is approximately 5715 years, which means if you began with a certain amount of \r{\({ }^{14}C\)}, after 5715 years, only half of it would remain. The concept of half-life is particularly useful because it provides a proportional decay rate which allows scientists to calculate the age of specimens regardless of the original quantity of the isotope present.