Problem 49 - Thomas Calculus

Question

Lascaux Cave paintings Prehistoric cave paintings of animals were found in the Lascaux Cave in France in \(1940 .\) Scientific analysis revealed that only 15\(\%\) of the original carbon-14 in the paintings remained. What is an estimate of the age of the paintings?

Step-by-Step Solution

Answer

The paintings are approximately 14,788 years old.

Step 1: Understanding the Concept of Carbon-14 Dating

Carbon-14 dating is a method used to determine the age of an artifact by measuring the remaining amount of carbon-14, a radioactive isotope. The half-life of carbon-14, the time it takes for half of the isotope to decay, is approximately 5730 years.

Step 2: Setting Up the Carbon-14 Decay Equation

The decay of carbon-14 can be modeled by the equation \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount of carbon-14 remaining at time \( t \), \( N_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is time. For this problem, \( N(t)/N_0 = 0.15 \).

Step 3: Calculating the Decay Constant

The decay constant \( k \) can be calculated using the half-life formula: \( k = \frac{\ln(2)}{5730} \approx 0.000121 \) per year.

Step 4: Solving for Time \( t \)

Using the equation \( N(t) = N_0 e^{-kt} \) and substituting \( N(t)/N_0 = 0.15 \), we have: \( 0.15 = e^{-kt} \). Taking the natural logarithm of both sides gives \( ln(0.15) = -kt \). Substitute \( k \approx 0.000121 \) to find \( t \): \( t = \frac{\ln(0.15)}{-0.000121} \approx 14788 \) years.

Key Concepts

Radioactive Isotope

When we talk about carbon-14 in the context of dating artifacts, we're diving into the world of radioactive isotopes. A radioactive isotope is a variant of an element that has an unstable nucleus, leading to radioactive decay over time.
Carbon-14 is an isotope of carbon, which means it has the same number of protons (6) as a typical carbon atom but a different number of neutrons, giving it an atomic mass of 14. This instability is what makes carbon-14 so special for dating ancient objects.

• As carbon-14 decays, it emits radiation and transforms into a different element, nitrogen-14.
• This decay process is consistent and measurable, which is fundamental to the method of carbon dating.
• Radioactive isotopes are used not only in dating but also in medical imaging and treatment, showcasing their versatility.

Understanding these isotopes helps us unlock the age of ancient items by measuring how much carbon-14 remains.

Half-Life

The concept of half-life is crucial to understanding radioactive decay. The half-life of a radioactive isotope is the time it takes for half of the isotope to decay into another element.
For carbon-14, this half-life is approximately 5730 years. This means that if you start with a specific amount of carbon-14, only half of that amount will remain after 5730 years.

• Knowing the half-life helps scientists calculate the remaining amount of a radioactive isotope at any given time.
• It provides a predictable measure of decay that is independent of environmental conditions.
• This constancy is what allows archaeologists and geologists to date ancient artifacts accurately.

By utilizing the half-life, we can backtrack to determine how old an artifact, such as the Lascaux Cave paintings, might be.

Decay Equation

The decay of radioactive substances can be mathematically represented using the decay equation. For carbon-14, this is expressed as \[ N(t) = N_0 e^{-kt} \] This equation allows us to model the exponential nature of radioactive decay.
Here, \( N(t) \) represents the amount of carbon-14 remaining at any given time \( t \), \( N_0 \) is the initial amount of carbon-14, \( k \) is the decay constant, and \( t \) is the time.

• The decay constant \( k \) quantifies the rate at which radioactive decay occurs and is specific to each isotope.
• The equation helps determine the time by relating the initial and remaining quantities of the isotope.
• It's a fundamental tool in carbon-14 dating, providing a mathematical framework to solve dating problems.

This equation is central to finding answers like the age of the paintings in the Lascaux Cave.

Natural Logarithm

The natural logarithm (\( \ln \)) plays a significant role in solving the decay equation. It is a type of logarithm that uses Euler's number \( e \) (approximately 2.718) as the base.
In the context of carbon-14 dating, taking the natural logarithm helps isolate and solve for the time variable \( t \) in the decay equation.

• By applying \( \ln \), you can transform the exponential equation into a linear one, making it easier to solve.
• The natural logarithm allows for the calculation of time based on known amounts of the radioactive isotope.
• It simplifies complex equations in various scientific and mathematical applications beyond just carbon dating.

Understanding the natural logarithm is vital for interpreting the results in decay-related problems, enabling us to decode the timeline of ancient artifacts precisely.