Problem 15 - College Algebra

Question

An artifact originally had 16 grams of carbon-l4 present. The decay model \(\bar{A}-16 e^{-9000121}\) 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 5715 years?

Step-by-Step Solution

Answer

The amount of carbon-14 left after 5715 years is calculated as approximately 8 grams.

Step 1: Understand the decay model

The given decay model is \(\bar{A}=16 e^{-0.000121t}\), where \(\bar{A}\) is the amount of carbon-14 left after \(t\) years.

Step 2: Substitute the value of 't'

In this exercise, we need to calculate the amount of carbon-14 left after 5715 years, so substitute \(t=5715\) in the decay model.

Step 3: Calculate \(\bar{A}\)

After substituting the value of \(t\) in the decay model, the equation becomes \(\bar{A}=16 e^{-0.000121 \times 5715}\). Calculating this will give us the amount of carbon-14 after 5715 years.

Key Concepts

Exponential Decay

Exponential decay is a process whereby the quantity of a substance decreases at a rate proportional to its current value. For substances like carbon-14, this process is described by the equation \( A(t) = A_0 e^{-kt} \), where:\
\

• \( A(t) \) is the amount of substance at time \( t \),
• \( A_0 \) is the initial amount of substance,
• \( e \) is the base of the natural logarithm,
• \( k \) is the decay constant, specific to the substance,
• \( t \) is the time elapsed.

In our case, the initial amount of carbon-14 \( A_0 \) is 16 grams, and the decay constant \( k \) is \( 0.000121 \) per year. Understanding the nature of exponential decay is crucial as it helps us predict how much of a substance will remain after a given period, which is fundamental in fields such as archaeology, geology, and environmental science for dating samples and understanding natural processes.

Radioactive Dating

Radioactive dating is a technique used to date materials like rocks or carbon, in which trace radioactive impurities were selectively incorporated when they formed. It relies on the principle of radioactive decay. For dating organic matter, scientists use isotopes like carbon-14, which is present in all living organisms. As living organisms take in carbon dioxide, a constant ratio of carbon-14 to carbon-12 is maintained while they are alive. Once the organism dies, it stops exchanging carbon with its environment, and the carbon-14 it contains begins to decrease through radioactive decay.To determine the age of an artifact or a sample, scientists measure the amount of carbon-14 present and use the decay rate to calculate back to the time when the organism died. This process gives us great insights into the historical and environmental record. The challenge is in accurately measuring very small changes in carbon-14 concentration over time, as this requires highly sensitive instruments and careful calibration.

Algebraic Modeling

Algebraic modeling involves translating real-world situations into mathematical models using algebraic expressions, equations or functions. In the context of carbon-14 decay, the algebraic model allows us to represent the exponential decrease of carbon-14 in an artifact over time. The given model in the exercise, \( \bar{A}=16 e^{-0.000121t} \), encapsulates all the factors affecting the decay process into a single expression.By plugging in different values of \( t \), the time since the organism's death, we can calculate the remaining amount of carbon-14, denoted by \( \bar{A} \). This approach demonstrates the power of algebra in solving real-life problems—by adopting a mathematical view of a process, we can make predictions, analyze trends, and derive meaningful insights about phenomena in the world around us. Algebraic modeling is an essential skill in various scientific disciplines, especially when dealing with processes that are governed by complex underlying principles.