Problem 31 - 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. To the nearest day, how long will it take for half of the Iodine-125 to decay?

Step-by-Step Solution

Answer

It will take approximately 60 days for half of the Iodine-125 to decay.

Step 1: Understanding the Problem

We are given an initial amount of Iodine-125 (0.5 grams) and a daily decay rate of 1.15%. We need to calculate the number of days it will take for this amount to decrease to half its initial value.

Step 2: Defining the Exponential Decay Formula

The exponential decay of a substance is given by the formula: \[ N(t) = N_0 imes (1 - r)^t \] where \( N_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time in days. We need to solve this equation for \( t \) when \( N(t) = 0.25 \) grams.

Step 3: Setting Up the Equation

Set \( N(t) = 0.25 \), \( N_0 = 0.5 \), and \( r = 0.0115 \) (since 1.15% as a decimal is 0.0115). Substitute these values into the formula:\[ 0.25 = 0.5 imes (1 - 0.0115)^t \]

Step 4: Isolating the Exponential Term

Divide both sides of the equation by 0.5 to isolate the exponential term:\[ 0.5 = (1 - 0.0115)^t \]

Step 5: Applying the Logarithm

To solve for \( t \), take the natural logarithm of both sides:\[ \ln(0.5) = \ln((1 - 0.0115)^t) \] By the property of logarithms, this becomes:\[ \ln(0.5) = t \cdot \ln(1 - 0.0115) \]

Step 6: Solving for Time \( t \)

Rearrange the equation to solve for \( t \): \[ t = \frac{\ln(0.5)}{\ln(1 - 0.0115)} \]Calculate the values using approximation:\[ t \approx \frac{-0.6931}{-0.01157} \approx 59.87 \]

Step 7: Rounding to the Nearest Day

Round 59.87 to the nearest whole number to obtain the final answer: 60 days.

Key Concepts

half-life

The half-life of a substance is the length of time it takes for half of the initial amount of a substance to decay or decrease to half its original amount. This concept is crucial for understanding the rate at which radioactive isotopes like Iodine-125 decay. Half-life is related to exponential decay and provides a practical measure of how long a radioactive isotope remains active. In simpler terms:

• If you start with 0.5 grams of a substance, by the end of its half-life, you'd expect to have 0.25 grams left.
• This same principle applies to each subsequent period of the half-life, meaning it continuously halves over equal time intervals.

Using the exponential decay formula, we see how half-life helps depict an exponential change: If you're given the decay rate (like 1.15% per day for Iodine-125), you can use the half-life to quickly assess how fast or slow the process is. The half-life provides a snapshot of this continuous exponential process by breaking it into easier-to-understand steps.

natural logarithm

The natural logarithm, denoted as ln, is a mathematical function that is heavily used in processes involving growth or decay, such as radioactive decay. It is necessary when isolating the variable time, which involves solving an exponent. In the scenario of Iodine-125 decay:

• The natural logarithm helps to turn complex exponential equations into linear ones, making it easier to solve for unknown variables like time (\(t\)).
• In our solution, we have \(\ln(0.5) = t \cdot \ln(1 - 0.0115)\), where we use ln to simplify multiplication across an exponent.

The natural logarithm is especially useful here because:

• It aids in figuring out time periods for which something changes rapidly and in small increments, such as radioactive decay.
• Using ln is straightforward as \(\ln(e^{x}) = x\), allowing simplicity when solving for \(t\).

Essentially, natural logarithms are the backbone of converting exponential data into manageable arithmetic, enabling us to derive timelines for processes like the decay of Iodine-125.

Iodine-125 decay

Iodine-125 (\(^125\text{I}\)) is a radioactive isotope commonly used in medical treatments, especially in radiation therapy for cancers. Understanding its decay process is crucial for safe and effective medical application. The decay of \(^125\text{I}\) is characterized by its specific rate of decay, which is given here as 1.15% per day.

• Exponential decay formula: \(N(t) = N_0 \times (1 - r)^t\), where \(r = 0.0115\) (decay rate).
• This formula helps practitioners calculate how rapidly the isotope’s activity decreases, guiding dosage and timing of treatments.

Iodine-125 decay is entirely predictable using these equations and concepts:

• After approximately 60 days (the half-life calculated here), the amount of \(^125\text{I}\) would have reduced to half, indicating clear predictable decay patterns.
• This predictability is indispensable in scenarios requiring precise drug delivery or radiation tactics.

Grasping the decay rate and the corresponding math ensures that applications involving \(^125\text{I}\) are not just effective but also safe.