Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.)

Step-by-Step Solution

Answer

The artifact is approximately 4432 years old.

Step 1: Understanding Radioactive Decay

Radioactive decay is a process where a quantity decreases at a rate proportional to its current value. The half-life of a substance is the time required for the quantity to reduce to half its initial value.

Step 2: Writing the Exponential Decay Formula

The formula for exponential decay is \( N(t) = N_0 \times e^{-kt} \) where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time. For carbon-14, we know its half-life is 5730 years.

Step 3: Calculating the Decay Constant

The decay constant \( k \) is calculated using the formula \( k = \frac{\ln 2}{\text{half-life}} \). So, \( k = \frac{\ln 2}{5730} \approx 0.000121 \text{ year}^{-1} \).

Step 4: Applying the Decay Formula

We want \( N(t) = 0.60 N_0 \). Using \( N(t) = N_0 \times e^{-kt} \), substitute the values: \( 0.60 N_0 = N_0 \times e^{-0.000121t} \). This simplifies to \( e^{-0.000121t} = 0.60 \).

Step 5: Solving for Time \( t \)

Take the natural logarithm on both sides: \( -0.000121t = \ln(0.60) \). Solving for \( t \) yields \( t = \frac{\ln(0.60)}{-0.000121} \approx 4432 \).

Step 6: Rounding to the Nearest Year

Finally, round the calculated \( t \approx 4432 \) to the nearest year. Therefore, the artifact is approximately 4432 years old.

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics that explains the process by which unstable atomic nuclei lose energy. This results in the emission of radiation. Radioactive substances transform over time into more stable forms by releasing particles or energy. One intriguing aspect of radioactive decay is its random nature, meaning it is impossible to predict exactly when a particular atom will decay.
Understanding how radioactive decay works is essential in numerous scientific fields. For example, in medicine, radioactive isotopes are used for both diagnosis and treatment. In archeology and geology, radioactive decay helps us date artifacts and fossil remains.
Key characteristics of radioactive decay include:

• It is an exponential process, meaning the rate of decay is proportional to the quantity of substance present.
• It is described mathematically using decay equations, which allow us to predict how much of a substance remains after a period of time.

The decay process is described by the decay constant, which helps measure how quickly a substance decays.

Half-life Calculations

The half-life is a crucial concept that describes the time required for half of a radioactive substance to decay. It is a measure of the stability of a radioactive isotope, with a longer half-life indicating greater stability. For example, carbon-14 has a half-life of 5730 years, which means that after this period, half of a given amount of carbon-14 will have decayed.
To calculate the remaining quantity of a substance after a certain period, scientists use the half-life in conjunction with decay formulas. This application is vital in fields such as archeology, where it helps date ancient artifacts.One way to calculate half-life is by using the exponential decay formula:

• The formula for decay: \[ N(t) = N_0 \times e^{-kt }\]
• From the formula, the decay constant \( k \) can be determined by \[ k = \frac{\ln(2)}{\text{Half-life}} \] where \( \ln(2) \approx 0.693 \) and
• A known half-life allows for precise calculations of how old a sample might be or how much of it remains over time.

Decay Constant

The decay constant, often represented by the symbol \( k \), is a key part of understanding radioactive decay. This value provides a rate at which a radioactive substance decays, and it is unique to each isotope. The larger the decay constant, the faster the substance decays.

• The decay constant connects directly to the half-life of an isotope. For example, it can be calculated using the formula: \[ k = \frac{\ln(2)}{\text{half-life}} \] where \( \ln(2) \approx 0.693 \).
• Once \( k \) is known, it can be used in the exponential decay formula to calculate how much of a substance remains after a given period.

Understanding \( k \) helps answer essential questions in both scientific research and practical applications, such as determining the age of archeological finds or planning medical treatments.

Carbon-14 Dating

Carbon-14 dating is a scientific method used to determine the age of an organic artifact. This technique relies on the radioactive decay of carbon-14, a naturally occurring isotope found in living organisms. Once an organism dies, the carbon-14 it contains decays at a known rate, allowing scientists to use this information to estimate the time since death.
Carbon-14 dating is particularly useful for dating material up to about 50,000 years old. This covers a range applicable to archaeology, paleontology, and geology. Here's how it typically works:

• A sample is tested to determine the amount of carbon-14 remaining compared to the amount present when the organism was alive.
• By knowing the half-life of carbon-14 (5730 years), scientists can use decay formulas to calculate the time elapsed since the organism died.
• This technique provides essential insights into past civilizations, environmental changes, and ecological dynamics.
• It is also pivotal in verifying and validating findings in other scientific areas.

Through carbon-14 dating, researchers have been able to establish timelines for prehistoric events and historical artifacts, contributing to our understanding of human history and the natural world.