Problem 19 - College Algebra

Question

Use the exponential decay model for carbon- \(14, A-A_{1} e^{-0,0001211}\) to solve Exercises \(19-20 .\) Prehistoric cave paintings were discovered in a cave in France. The paint contained \(15 \%\) of the original carbon- 14 . Estimate the age of the paintings.

Step-by-Step Solution

Answer

The estimated age of the cave paintings is about 18167 years.

Step 1: Understand the problem and identify the known and unknown values

We know that the remaining percentage of carbon-14 in the cave paintings is 15%, so \(A_1 = 0.15A\). We wish to find the time \(t\) in years. We will use the decay model equation to solve for \(t\).

Step 2: Substitute the given values into the decay model equation

Putting the given values into the equation, we get \(0 = A - 0.15A e^{-0.000121t}\). We can simplify this equation by dividing every term by \(A\), so we get \(0 = 1 - 0.15 e^{-0.000121t}\).

Step 3: Solve for \(t\)

Now we isolate \(e^{-0.000121t}\) by subtracting 1 from both sides and dividing by -0.15. Doing so, we get \(e^{-0.000121t} = \frac{1}{0.15} - 1\). Now, to solve for \(t\), we use the property of natural logarithms \(e^x = y\) implies \(x = \ln (y)\). Hence, applying the natural logarithm to both sides, we get \(-0.000121t = \ln(\frac{1}{0.15} - 1)\). Finally, we can solve for \(t\) by dividing both sides by -0.000121.

Step 4: Compute the result

Performing the above calculation, the estimated age of the paintings comes out to be approximately \(t = 18167\) years.

Key Concepts

Carbon-14 Dating

Carbon-14 dating is a fascinating technique used by archaeologists and scientists to determine the age of ancient organic materials. This method takes advantage of the natural decay process of carbon-14, a radioactive isotope of carbon. Carbon-14 is continuously formed in the atmosphere and is absorbed by living organisms. When an organism dies, it stops absorbing carbon-14, and the isotope begins to decay.
Understanding the decay process is crucial to dating ancient objects accurately. Carbon-14 decays at a steady rate, which is described using the concept of a half-life—the time it takes for half of the radioactive isotope to decay. For carbon-14, this half-life is approximately 5730 years, making it useful for dating things that are thousands of years old.
Through carbon-14 dating, we can estimate the age of prehistoric artifacts, like cave paintings, by measuring the remaining carbon-14 in a sample. If a sample shows only 15% of the original carbon-14, as in the given problem, it has undergone significant decay from its initial state. By using mathematical models, we can convert this percentage into an estimated age.

Exponential Functions

Exponential functions are a cornerstone of mathematical models that describe natural phenomena such as radioactive decay and population growth. At its core, an exponential function can be represented as \(y = Ae^{kt}\), where \(A\) is the initial quantity, \(e\) is the base of natural logarithms, \(k\) is the decay or growth rate, and \(t\) is time.
When dealing with exponential decay, like carbon-14 dating, the rate \(k\) is negative. This makes the function decrease over time, mirroring the reduction in the amount of carbon-14 in a sample. The equation provided in the step-by-step solution, \(0 = 1 - 0.15 e^{-0.000121t}\), is derived directly from an exponential decay model.

• "\(A\)" represents the original quantity of carbon-14.
• "0.15" reflects the remaining percentage of carbon-14.
• "\(-0.000121\)" is the decay rate specific to carbon-14.
• "\(t\)" is what we're solving for—time.

By manipulating the exponential decay equation, we can solve for \(t\), providing an estimation of the artifact's age.

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions. Simply put, while exponential functions explain how quantities grow or decay, logarithmic functions help us find the time it takes for a certain level of growth or decay. In the context of carbon-14 dating, they allow us to solve for "t," the age of the sample.
In the solution steps for the exercise, the natural logarithm function, represented as \(\ln\), was crucial. After rearranging the exponential decay formula to isolate \(e^{-0.000121t}\), we applied the logarithm to both sides of the equation:

\[-0.000121t = \ln\left(\frac{1}{0.15} - 1\right)\]
This step is key as it simplifies the problem, making it possible to find the value of \(t\) by solving:

• Apply \(\ln\) to both sides to handle the \(e\) term.
• Isolate \(t\) by dividing by the decay rate \(-0.000121\).

This process transforms the seemingly complex exponential decay model into a straightforward calculation, enabling us to estimate the time period a sample has been decaying, effectively revealing its age.