Problem 14 - College Algebra

Question

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbonI 4 present, \(A,\) in grams, after \(t\) years. How many grams of carbon-14 will be present after \(11,430\) years?

Step-by-Step Solution

Answer

Calculate \(A=16 e^{-0.0001211 * 11430}\) using a calculator to get the remaining amount of the carbon-14 after 11,430 years.

Step 1: Understand the Decay Formula

The decay formula given is \(A=16 e^{-0.0001211t}\). In this formula, A will give us the amount of carbon-14 left after t years. 16 represents the original amount of carbon-14, while \(-0.0001211\) is the decay rate.

Step 2: Insert the given value into the Formula

The problem asks the amount of carbon-14 left after \(11,430\) years. So, let's replace \(t\) in our formula with \(11,430\). We now have \(A=16 e^{-0.0001211 * 11430}\).

Step 3: Calculation

Next, take the exponent of \(-0.0001211 * 11430\) with the help of a calculator. After getting the exponent value, multiply it with 16 to find A, the amount of carbon-14 left after \(11,430\) years.

Key Concepts

Carbon-14 Dating

Carbon-14 dating is an essential method used in archaeology and other disciplines to determine the age of once-living materials. This technique relies on the properties of carbon-14, a radioactive isotope of carbon. Living organisms constantly exchange carbon with their environment, keeping the level of carbon-14 in almost perfect balance. However, when they die, the uptake of carbon stops, and the carbon-14 in their bodies begins to decay at a known rate. By measuring the remaining amount of carbon-14 in a sample, scientists can estimate how long it has been since the organism died. This is especially useful when dating ancient artifacts or organic remains, as it provides a way to approximate their age in years. The calculation involves understanding the remaining carbon-14 and the original amount that decayed over time.

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by radiation. In the context of carbon-14 dating, it refers to the transformation of carbon-14 to nitrogen-14 through a beta decay process. A key characteristic of radioactive decay is that it is exponential, meaning the rate of decay is proportional to the amount of substance remaining. Carbon-14 has a half-life of approximately 5730 years, which means that every 5730 years, half of the carbon-14 in a sample will have decayed. This predictable decay pattern forms the basis of carbon-14 dating, allowing scientists to calculate the age of carbon-bearing materials. During this decay, no external influences affect the rate, making it a reliable timekeeping method.

Decay Rate

The decay rate is a crucial component in the process of radioactive decay and aids in determining how quickly a radioactive substance transforms. In the given exercise, the decay rate of carbon-14 is represented by the constant (-0.0001211). Role of Decay Rate:

• The decay rate in the formula tells us how quickly carbon-14 is breaking down over time.
• It is a negative number, indicating a reduction of carbon-14 over time.
• Each radioactive element has a unique decay rate, which determines its half-life.

Understanding the decay rate enables scientists to create models that can accurately predict the amount of a radioactive substance present after a specific number of years. This rate is used in calculations to estimate the time that has elapsed since an organism that contained carbon-14 was last alive.

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the carbon-14 decay model, the exponential function is utilized to describe how the amount of carbon-14 decreases over time. The formula used, \(A = 16 \times e^{-0.0001211 \times t}\), is a type of exponential function.About Exponential Functions:

• They can model a variety of real-world processes, including radioactive decay.
• The base \(e\), approximately equal to 2.718, is a constant that represents the natural exponential function often used in continuous growth or decay models.
• The exponent in the function includes both the decay rate and the time variable, illustrating a diminishing function as time progresses.

These functions are powerful in accurately predicting changes over time, whether it's population growth, interest compounding, or, as in our case, the decay of carbon-14 isotopes.