Problem 32 - College Algebra

Question

Use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after \(t\) days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

Step-by-Step Solution

Answer

The amount of Iodine-125 remaining after 60 days is approximately 0.3 grams.

Step 1: Understand the decay process

Iodine-125 decays exponentially, meaning it decreases by a constant percentage over time. The decay rate given is 1.15% per day.

Step 2: Write the exponential decay formula

The general formula for exponential decay is given by \( A(t) = A_0 \times e^{-kt} \), where \( A_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is time. To find \( k \), we use the decay rate.

Step 3: Calculate the decay constant

The daily decay rate is 1.15%, which means \( k = \frac{1.15}{100} = 0.0115 \) per day. Thus, the decay model becomes \( A(t) = 0.5 \times e^{-0.0115t} \).

Step 4: Calculate the amount after 60 days

Substitute \( t = 60 \) into the exponential decay model: \( A(60) = 0.5 \times e^{-0.0115 \times 60} \).

Step 5: Compute the result

Calculate \( A(60) = 0.5 \times e^{-0.69} \approx 0.5 \times 0.501 \approx 0.2505 \). Round it to the nearest tenth, so the amount of Iodine-125 remaining is approximately 0.3 grams.

Key Concepts

Exponential Model

In the context of exponential decay, the model that predicts how a quantity decreases over time is called an exponential model. This type of model is described mathematically through an exponential function. When dealing with decay, the function ensures that the quantity diminishes by a consistent percentage over each unit of time.

The general exponential decay formula is expressed as \[A(t) = A_0 \times e^{-kt}\]where:

• \(A(t)\) represents the amount remaining after time \(t\).
• \(A_0\) is the initial quantity.
• \(e\) is the base of the natural logarithm, approximately equal to 2.718.
• \(k\) is the decay constant, which determines the rate of decay.
• \(t\) is the time that has elapsed.

In the given exercise, we see this model used to describe the decay of iodine-125 in a tumor. The decay constant \(k\) accounts for the daily decrease in iodine-125 due to its radioactive properties.

Decay Constant

The decay constant is a crucial part of the exponential decay model. It essentially captures the rate at which a substance diminishes over time. This constant allows the exponential model to reflect the observed decrease of the substance.

To find the decay constant \(k\), we convert a percentage decay rate into a decimal form because the model equation requires it. In this example:

• The given decay rate is a 1.15% daily decrease.
• This converts to a decay constant of \(k = \frac{1.15}{100} = 0.0115\) per day.

Once calculated, \(k\) is used in the equation \(A(t) = A_0 \times e^{-kt}\) to predict future amounts of the decaying substance, providing a reliable estimate depending on the amount of time elapsed.

Half-Life

Half-life is a term used to describe the time it takes for half of a substance to decay. It’s a concept often associated with radioactive materials, like Iodine-125. Knowing the half-life of a material allows scientists and researchers to understand how rapidly a substance is expected to decay, which is crucial in fields such as medicine and environmental science.

Interestingly, half-life is connected to the decay constant in the formula:\[T_{1/2} = \frac{\ln(2)}{k}\]where:

• \(T_{1/2}\) is the half-life period.
• \(\ln(2)\) is the natural logarithm of 2, approximately 0.693.
• \(k\) is the decay constant.

Understanding half-life helps in predicting how much of a substance will remain after a given period and is helpful in managing and disposing of radioactive materials safely.

Iodine-125

Iodine-125 is a radioactive isotope commonly used in medical applications, particularly in brachytherapy for cancer treatment. It emits gamma radiation, which is effective in targeting tumors while minimizing damage to surrounding healthy tissue.

In the given exercise, an initial amount of 0.5 gram of iodine-125 is used. The decay of iodine-125 is an example of exponential decay, where its amount decreases over time at a consistent rate, as shown in the problem using an exponential decay model.

This isotope has a relatively long half-life, which makes it practical for medical use, allowing enough time to administer and monitor treatment. Its behavior is predictable, which helps ensure patient safety and effective treatment outcomes.