Problem 16 - Algebra and Trigonometry

Question

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.000121 t}\) describes the amount of carbon- 14 present after \(t\) years. Use this model to solve Exercises \(15-16\) How many grams of carbon- 14 will be present in \(11,430\) years?

Step-by-Step Solution

Answer

To find the amount of Carbon-14 remaining after 11430 years, we substitute the time into the decay model and compute the output.

Step 1: Understanding the Model

The rate at which carbon-14 decays is expressed by the exponential decay model \(A=16 e^{-0.000121 t}\), where \(A\) is the amount of carbon-14 at time \(t\). The original quantity of carbon-14 is represented by 16 in this model.

Step 2: Substitute the Given Values into the Model

Next, we need to substitute the given time (t=11430 years) into the model equation: \(A=16 e^{-0.000121*11430}\).

Step 3: Problem Calculation

Upon simplifying the equation, we would find the amount of carbon-14 remaining after 11,430 years.

Key Concepts

Carbon-14 Decay

Carbon-14 decay is a fascinating natural phenomenon often used in archaeological and geological dating. It involves the transformation of carbon-14, a radioactive isotope, into nitrogen-14 over time. This transformation occurs through a spontaneous process called radioactive decay. Due to its predictable rate, scientists can use carbon-14 to estimate the age of ancient artifacts. This process is at the heart of radiocarbon dating, allowing us to unlock mysteries of the past.
For the decay rate, carbon-14 has a known half-life of approximately 5,730 years, meaning after this period, half of the original carbon-14 amount will have decayed. This steady rate of decay makes it ideal for dating items as old as 50,000 years. Using exponential decay functions, like the one in the original exercise, we can model and calculate how much carbon-14 remains after any given time period.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In biological and chemical processes like radioactive decay, exponential functions model rates of change.For carbon-14 decay, the function is formulated as an exponential decay model, represented by the equation:

• \( A = 16 e^{-0.000121 t} \)

Here, \( A \) represents the remaining amount of carbon-14, and \( e \) is the base of natural logarithms, approximately 2.718. The exponent \(-0.000121 t\) signifies the decay rate over time \(t\).Exponential decay functions describe processes that decrease at a rate proportional to their current value. This means as time increases, the carbon-14 decreases exponentially, resulting in a rapid reduction at first, which slows over time.

Mathematical Modeling

Mathematical modeling is a powerful tool that uses mathematics to represent real-world scenarios. It allows us to make predictions and understand complex systems simply and effectively. In our carbon-14 context, mathematical models help predict the remaining carbon-14 in artifacts over thousands of years.
The model provided in the exercise was:

• \( A = 16 e^{-0.000121 t} \)

This model includes specific parameters:

• The initial quantity of carbon-14, 16 grams.
• The decay constant, \( -0.000121 \), which determines the decay rate.
• The variable, \( t \), representing time in years.

By substituting known values into the model, such as \( t = 11430 \) years, we can solve for \( A \) to find how much carbon-14 remains. This demonstrates how mathematical modeling can transform abstract parameters into tangible predictions, making complex concepts more approachable.