Problem 20 - College Algebra

Question

Use the exponential decay model for carbon- \(14, A-A_{1} e^{-0,0001211}\) to solve Exercises \(19-20 .\) Skeletons were found at a construction site in San Francisco in \(1989 .\) The skeletons contained \(88 \%\) of the expected amount of carbon-14 found in a living person. In \(1989,\) how old were the skeletons?

Step-by-Step Solution

Answer

The skeletons found at the construction site in San Francisco in 1989 were approximately 1046 years old.

Step 1: Understand the problem and identify the given values

The problem is about the decay of carbon-14 in skeletons, using the exponential decay model. The given values are the percent of carbon-14 found in the skeletons (88%) and the decay rate of carbon-14 (-0.0001211). The exponential decay model is \(A = A_{1} e^{-0,0001211 t}\). We are asked to find the time \(t\), which represents the age of the skeletons.

Step 2: Substitute the given values into the formula

The basic form of the formula is \(A = A_{1} e^{-0,0001211 t}\). We know that \(A/A_{1} = 0.88\) from the amount of carbon-14 remaining in the skeleton. We can substitute this into the exponential decay model which will then look like this: \(0.88 = e^{-0,0001211 t}\).

Step 3: Solve for time \(t\)

We now have an equation with \(t\) as the only unknown. To find \(t\), we will first take natural logarithm on both sides of the equation. Due to the properties of logarithms, we can bring \(t\) in front of the logarithm. That gives us: \(\ln(0.88) = -0.0001211 t\). Finally, isolating \(t\) yields: \(t = \ln(0.88) / -0.0001211\).

Step 4: Calculate the age

Substituting the values into the formula and doing the arithmetic gives us the approximate age of the skeletons: \(t = \ln(0.88) / -0.0001211 = 1046\) years.

Key Concepts

carbon-14 dating

Carbon-14 dating is an essential method in archaeology and geology. It helps determine the age of ancient objects that contain organic material. Here's how it works.
All living things absorb carbon-14 from the atmosphere, maintaining a consistent level while alive. However, once they die, they stop absorbing it, and the existing carbon-14 starts to decay at a known rate.
Key aspects of carbon-14 dating:

• Carbon-14 is a radioisotope, meaning it is radioactive and decays over time.
• The decay happens at a predictable rate, described by the half-life, which is about 5,730 years for carbon-14.
• By measuring the remaining carbon-14 in a sample and comparing it to the expected level in a living organism, scientists can estimate the time since the organism's death.

Carbon-14 dating is especially useful for dating things that are up to about 50,000 years old. This makes it perfect for understanding human and animal history on Earth.

natural logarithm

The natural logarithm is a significant mathematical concept used in various fields, including carbon-14 dating. When working with exponential decay models, like in carbon dating, natural logarithms help us solve for unknowns.
What is a natural logarithm?

• It's the logarithm to the base e, where e ≈ 2.71828, which is a constant used in many areas of mathematics.
• In the context of decay problems, the natural logarithm allows us to "take apart" an exponential equation to make it solvable.

For example, if you have an equation like 0.88 = e^{-0.0001211 t}, applying the natural logarithm on both sides gives you:
\[ \ln(0.88) = -0.0001211 imes t \]
This makes it easy to isolate and solve for t, which is essential when estimating timescales in carbon-14 dating.

exponential functions

Exponential functions are a prevalent type of mathematical model that describes a variety of natural and man-made processes. They are especially crucial in understanding phenomena like growth and decay, such as in carbon-14 dating.
Characteristics:

• Exponential functions have the form \( f(x) = a \, e^{bx} \), where \( a \) is a constant, \( e \) is the base of natural logarithms, and \( b \) determines the rate of growth or decay.
• In the exponential decay model, the function describes a decline at a rate proportional to the current value. This is why they model phenomena like radioactive decay effectively.

With carbon-14 dating, we use an exponential decay model \( A = A_{1} e^{-0.0001211 t} \), where the rate of decay is given by \(-0.0001211\). By applying knowledge of exponential functions and logarithms, we can determine how long it has been since the organism or object stopped interacting with the atmosphere, giving us a means to date ancient artifacts.