Problem 85 - College Algebra and Calculus: An Applied Approach

Question

In Exercises \(83-86\), you are given the ratio of carbon atoms in a fossil. Use the information to estimate the age of the fossil. In living organic material, the ratio of radioactive carbon isotopes to the total number of carbon atoms is about 1 to \(10^{12}\). (See Example 2 in Section 10.1.) When organic material dies, its radioactive carbon isotopes begin to decay, with a half- life of about 5715 years. So, the ratio \(R\) of carbon isotopes to carbon- 14 atoms is modeled by \(R=10^{-12}\left(\frac{1}{2}\right)^{t / 5715}\), where \(t\) is the time (in years) and \(t=0\) represents the time when the organic material died. $$ R=0.22 \times 10^{-12} $$

Step-by-Step Solution

Answer

The fossil is approximately 9515 years old.

Step 1: Identify Given Ratio

First, identify the given ratio \(R\). From the problem, we know \(R = 0.22 \times 10^{-12}\)

Step 2: Apply Decay Model

Next, apply this ratio into the exponential decay model. Set \[ 0.22 \times 10^{-12} = 10^{-12} \times \left(\frac{1}{2}\right) ^{t / 5715} \]. Divide both sides by \(10^{-12}\) to simplify the equation and get \[ 0.22 = \left(\frac{1}{2}\right) ^{t / 5715} \].

Step 3: Simplify the equation

Now, we need to find time \(t\), so apply logarithmic rules. First take the natural log of both sides, which gives us: \[ \ln{0.22} = \ln{\left(\frac{1}{2}\right) ^{t / 5715}} \]. Using the property of natural logarithms, this further simplifies to: \[ \ln{0.22} = \frac{t}{5715} \ln{\frac{1}{2}} \].

Step 4: Solve for time

Lastly, solve the formula for time, \(t\), by multiplying both sides of the equation by 5715. We get \[ t = 5715 \times \frac{\ln{0.22}}{\ln{\frac{1}{2}}} \]. By using a calculator, \(t\) equals approximately 9515 years.

Key Concepts

Carbon Isotope Decay

The phenomenon of carbon isotope decay is fundamental to radiocarbon dating, a process used to determine the age of ancient organic material. Carbon is made up of isotopes, which are atoms of the same element with different numbers of neutrons. The most common isotopes of carbon are carbon-12, which is stable, and carbon-14, which is radioactive.

When a living organism dies, carbon-14 atoms begin to decay into nitrogen-14 atoms at a predictable rate. This decay is an example of radioactive decay, where the nucleus of an atom loses energy by emitting ionizing particles and radiation. Over time, the level of carbon-14 decreases while the level of carbon-12 remains constant, altering the ratio between the two.

To illustrate, initially in a living organism, the ratio of carbon-14 to total carbon atoms is roughly 1 to 1 trillion (1 to \(10^{12}\)). After the organism's death, the instantaneous decay of carbon-14 can be mathematically expressed, helping us to estimate when the organism passed away.

Exponential Decay Model

The exponential decay model is crucial in understanding the rate at which carbon-14 atoms decay. This model is based on the principle that the decay rate at any given time is directly proportional to the number of carbon-14 atoms present at that time. According to the model, the ratio \(R\) of the remaining carbon-14 isotopes in a fossil to the total number of carbon atoms decreases exponentially over time.

Mathematically, this model is described by the equation \( R = 10^{-12} \left(\frac{1}{2}\right)^{t / 5715} \), where \(t\) represents the time elapsed since the death of the organism and 5715 years is the half-life of carbon-14. A defining characteristic of the exponential decay model is that it produces a smooth, continuously decreasing curve when plotted on a graph, reflecting the steady reduction of the carbon-14 isotopes over time. This model allows for the precise calculation of the fossil's age by comparing the observed ratio \(R\) to the expected ratio in a living organism.

Half-Life Calculation

The concept of half-life is pivotal in the field of radioactive dating. A half-life is the time required for half of the radioactive isotopes in a sample to decay. For carbon-14, the half-life is approximately 5715 years. By knowing the half-life, scientists can calculate the age of a fossil based on the proportion of remaining carbon-14.

To determine the fossil's age, we calculate how many half-lives have passed since the organism died. This is done by comparing the current ratio of carbon isotopes to what we know was the ratio when the organism was alive. With the given half-life and the formula \( R=10^{-12}\left(\frac{1}{2}\right)^{t/5715} \), we solve for the time \(t\) that would yield the observed ratio \(R\). Logarithms help make this calculation manageable, allowing us to isolate \(t\) and estimate the time elapsed since the organism stopped exchanging carbon with the environment. As our example shows, we derived that \(t\) is roughly 9515 years, indicating the number of years since the organism's death.