Problem 404 - Precalculus

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of lodine- \(125,\) 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 lodine-125 that would remain in the tumor after 6 days to the nearest tenth of a gram.

Step-by-Step Solution

Answer

After 6 days, approximately 0.5 * 0.934 = 0.467 grams of Iodine-125 remain.

Step 1: Understand the Exponential Decay Formula

An exponential decay can be represented by the formula \( A(t) = A_0 \cdot e^{-kt} \), where \( A(t) \) is the amount remaining after time \( t \), \( A_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is time.

Step 2: Convert the Decay Rate to a Decay Constant

The problem states that the decay rate is 1.15% per day, which can be expressed as a decimal: 0.0115. The decay constant \( k \) is this value.

Step 3: Write the Exponential Decay Model

Now plug the given values into the decay formula: \( A(t) = 0.5 \cdot e^{-0.0115t} \). This represents the amount of Iodine-125 remaining after \( t \) days.

Step 4: Calculate the Amount after 6 Days

Substitute \( t = 6 \) into the model to find the amount of Iodine-125 remaining: \( A(6) = 0.5 \cdot e^{-0.0115 \cdot 6} \).

Step 5: Compute the Exponential Decay

First, calculate the exponent: \( -0.0115 \times 6 = -0.069 \). Then calculate the exponential \( e^{-0.069} \approx 0.9336 \).

Step 6: Final Calculation for Remaining Amount

Multiply by the initial amount: \( A(6) = 0.5 \times 0.9336 \approx 0.467 \) grams.

Key Concepts

Exponential Model

The exponential model helps us describe how quantities decrease over time if they are subject to exponential decay, just like Iodine-125 in a tumor. In an exponential decay model, we use the formula:

• \( A(t) = A_0 \cdot e^{-kt} \)
• \( A(t) \): the amount left after time \( t \)
• \( A_0 \): the initial quantity
• \( k \): the decay constant
• \( t \): the elapsed time

The exponential model effectively describes many real-life processes, like radioactive decay, population decline, or cooling processes. In our specific case with iodine, after injecting the initial amount into the tumor, the model allows us to predict how much will remain after a given time period. By understanding these variables and how they interact, we can create models that help predict outcomes under various circumstances. This is key in fields like medicine, where precise calculations can save lives.

Decay Constant

The decay constant \( k \) is a crucial element in the exponential decay formula. It quantifies the rate of decay per unit of time, letting us know how quickly or slowly the substance diminishes. In our exercise, the iodine isotope has a decay rate of 1.15% per day.
To find the decay constant, it is typically necessary to convert the percentage into a decimal. Therefore, the decay constant \( k \) is 0.0115 because 1.15% becomes 0.0115 as a decimal. This conversion is vital because the exponential model requires this in its calculations.
The decay constant provides a straightforward mechanism to understand and predict the behavior of the decaying substance over time, offering insights into how fast various processes occur.

Iodine-125

Iodine-125 is a radioactive isotope commonly used in medical applications, especially in brachytherapy -- a type of cancer treatment. This isotope, when introduced into the body, starts to decay, releasing radiation that can target and kill cancer cells.
Understanding the decay of Iodine-125 is essential for effective medical treatments. With a known half-life, the decay of this element can be predicted with high accuracy using exponential decay models. In the given exercise, the decay rate is 1.15% per day. With this knowledge, medical professionals can calculate how much Iodine-125 remains at any given time, helping optimize treatment plans.
By using precise models and constants, treatments that use Iodine-125 can be fine-tuned to maximize effectiveness while minimizing side effects, leading to better patient outcomes.