Problem 28 - Calculus Early Transcendentals

Question

The half-life of \(\mathrm{C}-14\) is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only \(77 \%\) of the C-14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has \(6.2 \%\) of the \(\mathrm{C}-14\) that it had when it was alive. Estimate when the wood was cut.

Step-by-Step Solution

Answer

Answer: The cloth was painted approximately 1,300 years ago and the wood was cut approximately 15,030 years ago.

Step 1: Understand the exponential decay formula

To determine the elapsed time since the cloth was painted, we first need to understand the exponential decay formula. The formula used for exponential decay is: $$N(t) = N_0 e^{-\lambda t},$$ where \(N(t)\) is the amount remaining at time t, \(N_0\) is the initial amount, \(\lambda\) is the decay constant, and \(t\) is time.

Step 2: Finding the decay constant

The decay constant, \(\lambda\), can be found using the formula: $$\lambda = \frac{\ln 2}{t_\frac{1}{2}},$$ where \(t_\frac{1}{2}\) is the half-life of the substance. In this case, the half-life of C-14 is 5730 years. So, we substitute the value into the equation to find the decay constant: $$\lambda = \frac{\ln 2}{5730}.$$

Step 3: Rewriting the exponential decay formula with values

Using the decay constant found in step 2, we can rewrite the exponential decay formula for this specific case: $$N(t) = N_0 e^{-\frac{\ln 2}{5730} t}.$$

Step 4: Finding the elapsed time

We are given that 77% of the C-14 in the dye remains. So, we can write the equation: $$0.77 = e^{-\frac{\ln 2}{5730} t}.$$ To solve for \(t\), take the natural logarithm of both sides: $$\ln 0.77 = -\frac{\ln 2}{5730} t.$$ Now, divide by the decay constant to find the elapsed time: $$t = \frac{\ln 0.77}{-\frac{\ln 2}{5730}} \approx 1300.46.$$ The cloth was painted approximately 1300 years ago. #b. Estimate when the wood was cut.#

Step 1: Use the exponential decay formula

We can use the same exponential decay formula as before: $$N(t) = N_0 e^{-\frac{\ln 2}{5730} t}.$$

Step 2: Finding the elapsed time

In this case, 6.2% of the C-14 in the wood remains. So, we can write the equation: $$0.062 = e^{-\frac{\ln 2}{5730} t}.$$ To solve for \(t\), take the natural logarithm of both sides: $$\ln 0.062 = -\frac{\ln 2}{5730} t.$$ Now, divide by the decay constant to find the elapsed time: $$t = \frac{\ln 0.062}{-\frac{\ln 2}{5730}} \approx 15030.18.$$ The wood was cut approximately 15,030 years ago.

Key Concepts

Radioactive Decay

Understanding the concept of radioactive decay is essential for many scientific applications, notably for dating artifacts in archaeology. This natural process occurs in unstable atomic nuclei that spontaneously transform into more stable forms over time by emitting radiation.

Imagine you have a collection of these unstable atoms. With each passing moment, some of them will decay, decreasing the overall number of radioactive atoms. The rate at which this happens is described mathematically by the exponential decay formula, reflecting the idea that the decay is proportional to the number of atoms remaining. This is crucial in determining how old an archaeological find is, as the amount of radioactive material, like Carbon-14, decreases predictably over time.

Half-Life

The half-life of a radioactive isotope is the time it takes for half of the original amount to decay. For instance, Carbon-14, a commonly analyzed isotope for dating organic material, has a half-life of approximately 5730 years.

This means if we start with 100 atoms of C-14, after 5730 years, only 50 would remain. After another 5730 years, 25 would be left, and so on. This predictable pattern allows scientists to estimate the age of materials that once were part of living organisms. By knowing the half-life and measuring the remaining amount of C-14, we can calculate how much time has passed since the organism stopped exchanging carbon with the environment, which typically happens upon death.

Natural Logarithm

A mathematical concept that often comes into play with radioactive decay is the natural logarithm, denoted as \(\ln\). The natural logarithm is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828.

In the context of radioactive decay, the natural logarithm is used to isolate the variable representing time when we have an exponential decay equation. As seen in the solutions above, by applying \(\ln\) to both sides of an exponential equation, we can solve for the time elapsed since the death of an organism or since an artifact was created.

C-14 Dating

Carbon-14 dating, also known as radiocarbon dating, is a method used to date materials that were once part of living organisms.

Every living organism contains the isotope C-14, which starts to decay at a steady rate once the organism dies. By measuring the remaining C-14 in a sample and using the known half-life of the isotope, we can calculate the time of death or last use. This technique has revolutionized the field of archaeology, allowing for precise dating of organic materials such as wood, textiles, and bones up to about 50,000 years old.

Archaeological Dating Methods

In archaeology, dating methods are vital for building a chronological framework to understand human history and prehistory.

In addition to radiocarbon dating, archaeologists use a variety of other techniques to date artifacts and sites. These may include dendrochronology, which uses the pattern of tree rings; thermoluminescence, which measures the light emitted from materials heated in the past; and potassium-argon dating, which is used for older geological samples. Each method has its own range of dates and types of materials it can date, making the selection of an appropriate technique a critical step in the archaeological analysis.